Adaptive monitoring of the progressive hedging penalty for reservoir systems management


Reservoir systems operations problems are in essence stochastic because of the uncertain nature of natural inflows. This leads to very large stochastic models that may not be easy to handle numerically. In this paper, we revisit the decomposition method developed by Rockafellar and Wets (Math Oper Res 119–147, 1991) by proposing new heuristics to initialize and dynamically adjust the penalty parameter of the augmented Lagrangian function on which this method is based. The heuristics are tested on multi-reservoir problems generated randomly and compared with the traditional strategy of setting the penalty parameter to a fixed value.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Benders, J.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4, 238–252 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Birge, J.: Decomposition and partitioning methods for multistage stochastic linear programs. Oper. Res. 33, 989–1007 (1985)

    Google Scholar 

  3. 3.

    Birge J, Louveaux F (1997) Introduction to Stochastic Programming. Springer-Verlag, New York (1997)

  4. 4.

    Blomvall, J., Lindberg, P.: A riccati-based primal interior point solver for multistage stochastic programming-extensions. Optimization Methods and Software 17(3), 383–407 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Dos Santos, M., Da Silva, E., Finardi, E., Gonçalves, R.: Practical aspects in solving the medium-term operation planning problem of hydrothermal power systems by using the progressive hedging method. International Journal of Electrical Power & Energy Systems 31(9), 546–552 (2009)

    Article  Google Scholar 

  6. 6.

    Dupačovà, J., Gröwe-Kuska, N., Römisch, W.: Scenario reduction in stochastic programming - an approach using probability metrics. Mathematical programming 95(3), 493–511 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Goncalves R., Finardi E., da Silva E.: Exploring the progressive hedging characteristics in the solution of the medium-term operation planning problem. In: Proceedings of 17th Power Systems Computation Conference, PSCC, Stockholm, Sweden (2011)

  8. 8.

    Kaut, M., Wallace, S.: Evaluation of scenario-generation methods for stochastic programming. Pacific Journal of Optimization 3(2), 257–271 (2007)

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Mulvey, J., Vladimirou, H.: Applying the progressive hedging algorithm to stochastic generalized networks. Annals of Operations Research 31(1), 399–424 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Pennanen T, Kallio M (2006) A splitting method for stochastic programs. Ann. Oper. Res. 142, 259–268 (2006)

    Google Scholar 

  11. 11.

    Pereira, M., Pinto, L.: Stochastic optimization of a multireservoir hydroelectric system: A decomposition approach. Water Resources Research 21(6), 779–792 (1985)

    Article  Google Scholar 

  12. 12.

    Pereira, M., Pinto, L.: Multi-stage stochastic optimization applied to energy planning. Mathematical Programming 52(1), 359–375 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Reis, F., Carvalho, P., Ferreira, L.: Reinforcement scheduling convergence in power systems transmission planning. IEEE Transactions on Power Systems 20(2), 1151–1157 (2005)

    Article  Google Scholar 

  14. 14.

    Rockafellar, R., Wets, J.R.: Scenarios and policy aggregation in optimization under uncertainty.Math. Oper. Res. 16, 119–147 (1991)

    Google Scholar 

  15. 15.

    Ruszczynski, A.: Some advances in decomposition methods for stochastic linear programming. Annals of Operations Research 85, 153–172 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Salinger, D., Rockafellar, R.: Dynamic splitting: An algorithm for deterministic and stochastic multiperiod optimization. Departement of Mathematics, University of Washington, Seattle, Working paper (2003)

  17. 17.

    Van Slyke R., Wets, R.: L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)

    Google Scholar 

  18. 18.

    Watson, J., Woodruff, D., Strip, D.: Progressive hedging innovations for a class of stochastic resource allocation problems. Working paper, Sandia National Laboratories, Alburquerque (2008)

Download references

Author information



Corresponding author

Correspondence to Luckny Zéphyr.

Additional information

This research was supported in part by the National Science and Engineering Research Council of Canada, under Grant 0105560.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zéphyr, L., Lang, P. & Lamond, B.F. Adaptive monitoring of the progressive hedging penalty for reservoir systems management. Energy Syst 5, 307–322 (2014).

Download citation


  • Reservoir systems operations
  • Stochastic programming models
  • Decomposition method
  • Heuristic
  • Progressive hedging algorithm