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Multi-usage hydropower single dam management: chance-constrained optimization and stochastic viability

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Abstract

We consider the management of a single hydroelectric dam, subject to uncertain inflows and electricity prices and to a so-called “tourism constraint”: the water storage level must be high enough during the tourist season with high enough probability. We cast the problem in the stochastic optimal control framework: we search at each time t the optimal control as a function of the available information at t. We lay out two approaches. First, we formulate a chance-constrained stochastic optimal control problem: we maximize the expected gain while guaranteeing a minimum storage level with a minimal prescribed probability level. Dualizing the chance constraint by a multiplier, we propose an iterative algorithm alternating additive dynamic programming and update of the multiplier value “à la Uzawa”. Our numerical results reveal that the random gain is very dispersed around its expected value; in particular, low gain values have a relatively high probability to materialize. This is why, to put emphasis on these low values, we outline a second approach. We propose a so-called stochastic viability approach that focuses on jointly guaranteeing a minimum gain and a minimum storage level during the tourist season. We solve the corresponding problem by multiplicative dynamic programming. To conclude, we discuss and compare the two approaches.

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Notes

  1. Even if it was not the case in the data provided by Electricité de France.

  2. The abbreviation w.r.t. stands for “with respect to”.

  3. Whereas it would correspond to the Decision-Hazard framework if \( \mathbf {U}_{t} \) were measurable w.r.t. \( \sigma \left( \mathbf {W}_{0}, \, \ldots , \, \mathbf {W}_{t-1} \right) \) (see [7]).

  4. The abbreviation s.t. stands for “such that”.

  5. The gradient step method for the dual minimization problem may be replaced by a more efficient method such as dichotomy, conjugate gradient or quasi-Newton.

  6. Otherwise, the multiplier goes to infinity with the iteration index, which means that Constraint (9f) is infeasible.

  7. Although this assumption is by no means required.

  8. See Remark 2 for the influence of these discretization choices on the quality of the solution.

  9. We could consider that the set in which \( \mathbf {S}_{t} \) takes its values might vary with respect to t. This would certainly reduce the algorithm running time, but it would not reduce it by orders of magnitude.

References

  1. Van Ackooij, W.: Chance Constrained Programming with applications in Energy Management. PhD thesis, École Centrale des Arts et Manufactures (2013)

  2. Van Ackooij, W., Zorgati, R., Henrion, R., Möller, A.: Joint chance-constrained programming for hydro reservoir management. Optim. Eng. 15, 509–531 (2014)

    MathSciNet  Google Scholar 

  3. Alais, J.C.: Risque et optimisation pour le management d’énergies: application à l’hydraulique. PhD thesis, Université Paris-Est (2013)

  4. Andrieu, L., Henrion, R., Römisch, W.: A model for dynamic chance constraints in hydro power reservoir management. Eur. J. Oper. Res. 207, 579–589 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charnes, A., Cooper, W.W.: Chance-constrained programming. Manag. Sci. 6, 73–79 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carpentier, P., Chancelier, J.-P., Cohen, G., De Lara, M., Girardeau, P.: Dynamic consistency for stochastic optimal control problems. Ann. Oper. Res. 200, 247–263 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carpentier, P., Chancelier, J.-P., Cohen, G., De Lara, M.: Stochastic multi-stage optimization. At the crossroads between discrete time stochastic control and stochastic programming. Springer, Berlin (2015)

    MATH  Google Scholar 

  8. De Lara, M., Doyen, L.: Sustainable management of natural resources: mathematical models and methods. Environmental science and engineering. Springer, Berlin (2008)

    Google Scholar 

  9. Dentcheva, D.: Optimization models with probabilistic constraints. In Lecture Notes on Stochastic Programming Modeling and Theory, pp. 87–153 (2009)

  10. Doyen, L., De Lara, M.: Stochastic viability and dynamic programming. Syst. Control Lett. 59(10), 629–634 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dupačová, J.: Stability and sensitivity analysis for stochastic programming. Ann. Oper. Res. 27(1), 115–142 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ekeland, I., Temam, R.: Convex analysis and variational problems. Studies in mathematics and its applications. North-Holland Pub Co., New York (1999)

    MATH  Google Scholar 

  13. Everett, H.: Generalized Lagrange multiplier method for solving problems of optimum allocation of resources. Oper. Res. 11, 399–417 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henrion, R.: On the connectedness of probabilistic constraint sets. J. Optim. Theory Appl. 112(3), 657–663 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Henrion, R.: A critical note on empirical (sample average, Monte Carlo) approximation of solutions to chance constrained programs. In:Homberg, D., Troltzsch, F. (eds.) System Modeling and Optimization. IFIP Advances in Information and Communication Technology, vol. 391,pp. 25–37. Springer, Hidelberg (2013)

  16. Henrion, R., Römisch, W.: Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints. Math. Program. 100(3), 589–611 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Infanger, G., Morton, D.P.: Cut sharing for multistage stochastic linear programs with interstage dependency. Math. Program. 75(2), 241–256 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maceira, M.E., Damazio, J.M.: The use of PAR(p) model in the stochastic dual dynamic programming optimization scheme used in the operation planning of the Brazilian hydropower system Operation Planning Studies of the Brazilian Generating System. In: IEEE 8th International Conference on Probabilistic Methods Applied to Power Systems,pp. 397–402, Ames, Iowa, (2004)

  20. Miller, L.B., Wagner, H.: Chance-constrained programming with joint constraints. Oper. Res. 13, 930–945 (1965)

    Article  MATH  Google Scholar 

  21. Ono, M., Kuwata Y., Balaram J.: Joint chance-constrained dynamic programming. In: 51st IEEE Conference on Decision and Control, pp. 1915–1922. Maui, Hawaii (2012)

  22. Prékopa, A., Szántai, T.: On optimal regulation of a storage level with application to the water level regulation of a lake. Europ. J. Oper. Res. 3(3), 175–189 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Prékopa, A.: Stochastic Programming. Mathematics and Its Applications. Kluwer Academic Publisher, Dordrecht (1995)

    MATH  Google Scholar 

  24. Prékopa, A.: Probabilistic programming. In: Ruszczynski, A., Shapiro, A. (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, vol. 10, pp. 267–351. Elsevier, Amsterdam (2003)

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Acknowledgments

The authors thank Electricité de France Research and Development for initiating this research through the CIFRE PhD funding of Jean-Christophe Alais and for supplying us with data.

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Authors and Affiliations

  1. Artelys, 12, rue du Quatre Septembre, Paris, 75002, France

    Jean-Christophe Alais

  2. UMA, ENSTA ParisTech, Université Paris-Saclay 828, Boulevard des Maréchaux, 91762, Palaiseau cedex, France

    Pierre Carpentier

  3. Université Paris-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455, Marne-la-Vallée, France

    Michel De Lara

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  1. Jean-Christophe Alais
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  2. Pierre Carpentier
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  3. Michel De Lara
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Correspondence to Pierre Carpentier.

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Alais, JC., Carpentier, P. & De Lara, M. Multi-usage hydropower single dam management: chance-constrained optimization and stochastic viability. Energy Syst 8, 7–30 (2017). https://doi.org/10.1007/s12667-015-0174-4

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