Management of a hydropower system via convex duality

  • Kristina Rognlien Dahl
Original Article


We consider a stochastic hydroelectric power plant management problem in discrete time with arbitrary scenario space. The inflow to the system is some stochastic process, representing the precipitation to each dam. The manager can control how much water to turbine from each dam at each time. She would like to choose this in a way which maximizes the total profit from the initial time 0 to some terminal time T. The total profit of the hydropower dam system depends on the price of electricity, which is also a stochastic process. The manager must take this price process into account when controlling the draining process. However, we assume that the manager only has partial information of how the price process is formed. She can observe the price, but not the underlying processes determining it. By using the conjugate duality framework, we derive a dual problem to the management problem. This dual problem turns out to be simple to solve in the case where the profit rate process is a martingale or submartingale with respect to the filtration modeling the information of the dam manager. In the case where we only consider a finite number of scenarios, solving the dual problem is computationally more efficient than the primal problem.


Stochastic control Hydropower management Conjugate duality Martingales 



The authors would like to thank two anonymous reviewers for their helpful comments which have significantly improved the paper.


  1. Aasgård EK, Andersen GS, Fleten S-E, Haugstved D (2014) Evaluating a stochastic programming based bidding model for a multireservoir system. IEEE Trans Power Syst 29:1748–1757. CrossRefGoogle Scholar
  2. Alais J-C, Capentier P, De Lara M (2017) Multi-usage hydropower single dam management: chance-constrained optimization and stochastic viability. Energy Syst 8:7–30. CrossRefGoogle Scholar
  3. Anderson EJ, Nash P (1987) Linear programming in infinite dimensional spaces: theory and applications. Wiley-Interscience Series in Discrete Mathematics and Optimization, HobokenzbMATHGoogle Scholar
  4. Apostolopoulou D, De Gréve Z, McCulloch M (2018) Robust optimization for hydroelectric system operation under uncertainty. IEEE Trans Power Syst 33:3337–3348. CrossRefGoogle Scholar
  5. Benth FE, Kallsen J, Meyer-Brandis T (2007) A non-Gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing. Appl Math Finance 14:153–169. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bierbrauer M, Menn C, Rachev ST, Trueck S (2007) Spot and derivative pricing in the EEX power market. J Bank Finance 31:3462–3485. CrossRefGoogle Scholar
  7. Chen Z, Forsyth PA (2008) Pricing hydroelectric power plants with/without operational restrictions: a stochastic control approach. Nonlinear Models in Mathematical Finance. Nova Science Publishers, New YorkGoogle Scholar
  8. Dahl KR (2017) A convex duality approach for pricing contingent claims under partial information and short selling constraints. Stoch Anal Appl 35:317–333. MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dahl KR, Stokkereit E (2016) Stochastic maximum principle with Lagrange multipliers and optimal consumption with Lévy wage. Afrika Matematika 27:555–572. MathSciNetCrossRefzbMATHGoogle Scholar
  10. Devolder O, Glineur F, Nesterov Y (2010) Solving infinite-dimensional optimization problems by polynomial approximation. Center for Research and Econometrics Discussion PaperGoogle Scholar
  11. Dohrman CR, Robinett RD (1999) Dynamic programming method for constrained discrete time optimal control. J Optim Theory Appl 101:259–283. MathSciNetCrossRefGoogle Scholar
  12. Gauvin C, Delage E, Gendreau M (2018) A successive linear programming algorithm with non-linear time series for the reservoir management problem. Comput Manag Sci 15:55–86. MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hamann A, Hug G, Rosinski S (2017) Real-Time optimization of the mid-Columbia hydropower system. IEEE Trans Power Syst 32:157–165. CrossRefGoogle Scholar
  14. Härtel P, Korpås M (2017) Aggregation methods for modelling hydropower and its implications for a highly decarbonised energy system in Europe. Energies 10:1842. CrossRefGoogle Scholar
  15. Huseby AB (2016) Optimizing energy production systems under uncertainty. In: Lesley Walls, Matthew Revie (eds) Risk, Reliability and Safety: innovating theory and practice: proceedings of ESREL 2016.
  16. Ji S, Zhou XY (2006) A maximum principle for stochastic optimal control with terminal state constraints and its applications. Commun Inf Syst 6:321–338MathSciNetzbMATHGoogle Scholar
  17. King AJ (2002) Duality and martingales: a stochastic programming perspective on contingent claims. Math Program Ser B 91:543–562. MathSciNetCrossRefzbMATHGoogle Scholar
  18. King AJ, Korf L (2001) Martingale pricing measures in incomplete markets via stochastic programming duality in the dual of L1 (preprint). University of Washington. Accessed 25 July 2017
  19. Löschenbrand M, Korpås M (2017) Hydro power reservoir aggregation via genetic algorithms. Energies.
  20. Niu S, Insley M (2016) An options pricing approach to ramping rate restrictions at hydro power plants. J Econ Dyn Control 63:25–52. MathSciNetCrossRefzbMATHGoogle Scholar
  21. Øksendal B (2000) Stochastic differential equations: an introduction with applications. Springer, BerlinzbMATHGoogle Scholar
  22. Pennanen T (2011) Convex duality in stochastic optimization and mathematical finance. Math Oper Res 36:340–362MathSciNetCrossRefzbMATHGoogle Scholar
  23. Pennanen T (2012) Introduction to convex optimization in financial markets. Math Program 134:157–186. MathSciNetCrossRefzbMATHGoogle Scholar
  24. Pennanen T, Perkkiö AP (2012) Stochastic programs without duality gaps. Math Program 136:91–110. MathSciNetCrossRefzbMATHGoogle Scholar
  25. Rambharat BR, Brockwell AE, Seppi DJ (2005) A threshold autoregressive model for wholesale electricity prices. J R Stat Soc Ser C (Appl Stat) 54:287–299MathSciNetCrossRefzbMATHGoogle Scholar
  26. Rockafellar RT (1972) Convex analysis. Princeton University Press, PrincetonGoogle Scholar
  27. Rockafellar RT (1974) Conjugate duality and optimization. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefzbMATHGoogle Scholar
  28. Rockafellar RT, Wets RJ-B (2004) Variational analysis. Springer, BerlinzbMATHGoogle Scholar
  29. Shayesteh E, Amelin M, Söder L (2016) Multi-station equivalents for short-term hydropower scheduling. IEEE Trans Power Syst 31:4616–4625. CrossRefGoogle Scholar
  30. Vanderbei RJ (1996) Linear programming: foundations and extensions. Kluwer Academic Publishers, BostonzbMATHGoogle Scholar
  31. Weron R (2014) Electricity price forecasting: a review of the state-of-the-art with a look into the future. Int J Forecast 30:1030CrossRefGoogle Scholar
  32. Weron R, Bierbrauer M, Truck S (2004) Modeling electricity prices: jump diffusion and regime switching. Physica A 336:39–48. CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway

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