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A Common Regularization for Three Reservoir Optimal Control Problems

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.

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References

  1. Arunkumar, S., Chon, K.: On optimal regulation policies for certain multi-reservoir systems. Oper. Res. 26, 551–562 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  5. Bauer, W., Gfrerer, H., Wacker, H.: Optimization strategies for hydro energy storage plants. Z. Oper.-Res. B 28, 103–131 (1984)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  7. Gfrerer, H.: Optimization of hydro energy storage plant problems by variational methods. Z. Oper.-Res. B 28, 87–101 (1984)

    MathSciNet  MATH  Google Scholar 

  8. Phu, H.X.: On the optimal control of a hydroelectric power plant. Syst. Control Lett. 8, 281–288 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MATH  Article  Google Scholar 

  10. Phu, H.X.: Optimal control of a hydroelectric power plant with unregulated spilling water. Syst. Control Lett. 10, 131–139 (1988)

    MATH  Article  Google Scholar 

  11. Chang, L.-C.: Guiding rational reservoir flood operation using penalty-type genetic algorithm. J. Hydrol. 354, 65–74 (2008)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MATH  Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  19. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923)

    MATH  Google Scholar 

  20. Tikhonov, A.N.: Solution of incorrect formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035–1038 (1963)

    Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  23. Phu, H.X.: Zur Lösung einer regulären Aufgabenklasse der optimalen Steuerung im Großen mittels Orientierungskurven. Optimization 18, 65–81 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  24. Phu, H.X.: Method of orienting curves for solving optimal control problems with state constraints. Numer. Funct. Anal. Optim. 12, 173–211 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  25. Phu, H.X., Dinh, N.: Some remarks on the method of orienting curves. Numer. Funct. Anal. Optim. 16, 755–763 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  26. Phu, H.X.: Zur Lösung eines Zermeloschen Navigationsproblems. Optimization 18, 225–236 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  27. Phu, H.X.: Ein konstruktives Lösungsverfahren für das Problem des Inpolygons kleinsten Umfangs von J. Steiner. Optimization 18, 349–359 (1987)

    MathSciNet  MATH  Article  Google Scholar 

  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)

    MathSciNet  MATH  Article  Google Scholar 

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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.

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Correspondence to H. X. Phu.

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Communicated by Nguyen Dong Yen.

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Phu, H.X., Bock, H.G. A Common Regularization for Three Reservoir Optimal Control Problems. J Optim Theory Appl 157, 199–228 (2013). https://doi.org/10.1007/s10957-012-0173-8

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  • DOI: https://doi.org/10.1007/s10957-012-0173-8

Keywords

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