Advertisement

A Common Regularization for Three Reservoir Optimal Control Problems

  • H. X. PhuEmail author
  • H. G. Bock
Article
  • 290 Downloads

Abstract

Four optimal control problems of reservoir release are investigated. The first problem is to minimize the peak release in order to prevent flood and to reduce the flood height. The second problem is to maximize the lowest release in order to ensure irrigation, water supply, shipping and environment downstream. The third problem is to minimize the flooding duration in order to reduce damage to goods, possessions, plants, levees, etc. It is shown that these three problems may possess infinitely many different optimal solutions, but they all have a common optimal solution, which is the unique optimal solution of the fourth problem. Since this unique optimal solution depends continuously on the input data, the fourth problem is well-posed and it can be considered as a common regularization of the three ill-posed problems.

Keywords

Optimal control Reservoir release Flood prevention Flood damage reduction Irrigation Water supply Ill-posedness Regularization 

Notes

Acknowledgements

This research was funded by Vietnam National Foundation for Science and Technology Development under grant number 101.02-2011.45. The authors thank the editors and the referees for their helpful comments.

References

  1. 1.
    Arunkumar, S., Chon, K.: On optimal regulation policies for certain multi-reservoir systems. Oper. Res. 26, 551–562 (1978) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Edirisinghe, N.C.P., Patterson, E.I., Saadouli, N.: Capacity planning model for a multipurpose water reservoir with target-priority operation. Ann. Oper. Res. 100, 273–303 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Keckler, W.G., Larson, R.E.: Dynamic programming applications to water resource system operation and planning. J. Math. Anal. Appl. 24, 80–109 (1968) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Sarma, P., Durlofsky, L.J., Aziz, K., Chen, W.H.: Efficient real-time reservoir management using adjoint-based optimal control and model updating. Comput. Geosci. 10, 3–36 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bauer, W., Gfrerer, H., Wacker, H.: Optimization strategies for hydro energy storage plants. Z. Oper.-Res. B 28, 103–131 (1984) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dinh, N., Phu, H.X.: The method of orienting curves and its application to an optimal control problem of hydroelectric power plants. Vietnam J. Math. 20, 40–53 (1992) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gfrerer, H.: Optimization of hydro energy storage plant problems by variational methods. Z. Oper.-Res. B 28, 87–101 (1984) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Phu, H.X.: On the optimal control of a hydroelectric power plant. Syst. Control Lett. 8, 281–288 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Phu, H.X.: On a linear optimal control problem of a system with circuit-free graph structure. Int. J. Control 48, 1867–1882 (1988) zbMATHCrossRefGoogle Scholar
  10. 10.
    Phu, H.X.: Optimal control of a hydroelectric power plant with unregulated spilling water. Syst. Control Lett. 10, 131–139 (1988) zbMATHCrossRefGoogle Scholar
  11. 11.
    Chang, L.-C.: Guiding rational reservoir flood operation using penalty-type genetic algorithm. J. Hydrol. 354, 65–74 (2008) CrossRefGoogle Scholar
  12. 12.
    Hsu, N.-S., Wei, C.-C.: A multipurpose reservoir real-time operation model for flood control using typhoon invasion. J. Hydrol. 336, 282–293 (2007) CrossRefGoogle Scholar
  13. 13.
    Karbowski, A.: Optimal control of single retention reservoir during flood: solution of deterministic, continuous-time problems. J. Optim. Theory Appl. 69, 55–81 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Pytlak, R., Malinowski, K.: Optimal scheduling of reservoir releases during flood: deterministic optimization problem, part 1. Procedure. J. Optim. Theory Appl. 61, 409–432 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Pytlak, R., Malinowski, K.: Optimal scheduling of reservoir releases during flood: deterministic optimization problem, part 2. Case study. J. Optim. Theory Appl. 61, 433–449 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chatterjee, B., Howitt, R.E., Sexton, R.J.: The optimal joint provision of water for irrigation and hydropower. J. Environ. Econ. Manag. 36, 295–313 (1998) zbMATHCrossRefGoogle Scholar
  17. 17.
    Elferchichi, A., Gharsallah, O., Nouiri, I., Lebdi, F., Lamaddalena, N.: The genetic algorithm approach for identifying the optimal operation of a multi-reservoirs on-demand irrigation system. Biosyst. Eng. 1002, 334–344 (2009) CrossRefGoogle Scholar
  18. 18.
    Umamahesh, N.V., Sreenivasulu, P.: Two-phase stochastic dynamic programming model for optimal operation of irrigation reservoir. Water Res. Manag. 11, 395–406 (1997) CrossRefGoogle Scholar
  19. 19.
    Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923) zbMATHGoogle Scholar
  20. 20.
    Tikhonov, A.N.: Solution of incorrect formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035–1038 (1963) Google Scholar
  21. 21.
    Dinh, N., Phu, H.X.: Solving a class of regular optimal control problems with state constraints by the method of orienting curves. Optimization 25, 231–247 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dinh, N., Phu, H.X.: Solving a class of optimal control problems which are linear in the control variable by the method of orienting curves. Acta Math. Vietnam. 17, 115–134 (1992) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Phu, H.X.: Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung im Großen mittels Orientierungskurven. Optimization 18, 65–81 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Phu, H.X.: Method of orienting curves for solving optimal control problems with state constraints. Numer. Funct. Anal. Optim. 12, 173–211 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Phu, H.X., Dinh, N.: Some remarks on the method of orienting curves. Numer. Funct. Anal. Optim. 16, 755–763 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Phu, H.X.: Zur Lösung eines Zermeloschen Navigationsproblems. Optimization 18, 225–236 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Phu, H.X.: Ein konstruktives Lösungsverfahren für das Problem des Inpolygons kleinsten Umfangs von J. Steiner. Optimization 18, 349–359 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Phu, H.X., Bock, H.G., Schlöder, J.: The method of orienting curves and its application for manipulator trajectory planning. Numer. Funct. Anal. Optim. 18, 213–225 (1997) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany

Personalised recommendations