Optimal control of electricity input given an uncertain demand

  • Simone GöttlichEmail author
  • Ralf Korn
  • Kerstin Lux
Original Article


We consider the problem of determining an optimal strategy for electricity injection that faces an uncertain power demand stream. This demand stream is modeled via an Ornstein–Uhlenbeck process with an additional jump component, whereas the power flow is represented by the linear transport equation. We analytically determine the optimal amount of power supply for different levels of available information and compare the results to each other. For numerical purposes, we reformulate the original problem in terms of the cost function such that classical optimization solvers can be directly applied. The computational results are illustrated for different scenarios.


Stochastic optimal control Jump diffusion processes Transport equation 

Mathematics Subject Classification

93E20 60H10 65C20 



The authors are grateful for the support of the German Research Foundation (DFG) within the Project “Novel models and control for networked problems: from discrete event to continuous dynamics” (GO1920/4-1) and the BMBF within the Project ENets.


  1. Aïd R, Campi L, Huu AN, Touzi N (2009) A structural risk-neutral model of electricity prices. Int J Theor Appl Financ 12:925–947MathSciNetCrossRefzbMATHGoogle Scholar
  2. Annunziato M, Borzì A (2013) A Fokker–Planck control framework for multidimensional stochastic processes. J Comput Appl Math 237:487–507MathSciNetCrossRefzbMATHGoogle Scholar
  3. Annunziato M, Borzì A (2018) A Fokker–Planck control framework for stochastic systems. EMS Surv Math Sci 5:65–98MathSciNetCrossRefzbMATHGoogle Scholar
  4. Applebaum D (2009) Lévy processes and stochastic calculus, vol. 116 of Cambridge studies in advanced mathematics, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  5. Barlow MT (2002) A diffusion model for electricity prices. Math Financ 12:287–298CrossRefzbMATHGoogle Scholar
  6. Benth F, Benth J, Koekebakker S (2008) Stochastic modelling of electricity and related markets, vol. 11 of advanced series on statistical science & applied probability. World Scientific Publishing Co. Pte. Ltd., HackensackzbMATHGoogle Scholar
  7. Breitenbach T, Annunziato M, Borzì A (2018) On the optimal control of a random walk with jumps and barriers. Methodol Comput Appl Probab 20:435–462MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gaviraghi B, Annunziato M, Borzì A (2017) A Fokker–Planck based approach to control jump processes. In: Ehrhardt M, Günther M, ter Maten EJW (eds) Novel methods in computational finance, vol. 25 of mathematics in industry. Springer, Cham, pp 423–439 Google Scholar
  9. Göttlich S, Herty M, Schillen P (2016) Electric transmission lines: control and numerical discretization. Optim Control Appl Methods 37:980–995MathSciNetCrossRefzbMATHGoogle Scholar
  10. Göttlich S, Teuber C (2018) Space mapping techniques for the optimal inflow control of transmission lines. Optim Methods Softw 33:120–139MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kiesel R, Schindlmayr G, Börger RH (2009) A two-factor model for the electricity forward market. Quant Financ 9:279–287MathSciNetCrossRefzbMATHGoogle Scholar
  12. Klenke A (2008) Probability theory: a comprehensive course. Springer, LondonCrossRefzbMATHGoogle Scholar
  13. Korn R, Korn E, Kroisandt G (2010) Monte Carlo methods and models in finance and insurance. Chapman & Hall/CRC Financial Mathematics Series, CRC Press, Boca RatonCrossRefzbMATHGoogle Scholar
  14. La Marca M, Armbruster D, Herty M, Ringhofer C (2010) Control of continuum models of production systems. IEEE Trans Automat Control 55:2511–2526MathSciNetCrossRefzbMATHGoogle Scholar
  15. LeVeque RJ (1990) Numerical methods for conservation laws, lectures in mathematics ETH Zürich. Birkhäuser Verlag, BaselCrossRefGoogle Scholar
  16. Lucia JJ, Schwartz ES (2002) Electricity prices and power derivatives: evidence from the Nordic power exchange. Rev Deriv Res 5:5–50CrossRefzbMATHGoogle Scholar
  17. Mikosch T (2009) Non-life insurance mathematics: an introduction with the Poisson process, 2nd edn. Universitext, Springer, BerlinCrossRefzbMATHGoogle Scholar
  18. Roy S, Annunziato M, Borzì A, Klingenberg C (2018) A Fokker–Planck approach to control collective motion. Comput Optim Appl 69:423–459MathSciNetCrossRefzbMATHGoogle Scholar
  19. Schwartz E, Smith JE (2000) Short-term variations and long-term dynamics in commodity prices. Manag Sci 46:893–911CrossRefGoogle Scholar
  20. Wagner A (2014) Residual demand modeling and application to electricity pricing. Energy J 35:45–73CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MannheimMannheimGermany
  2. 2.Department of MathematicsTU KaiserslauternKaiserslauternGermany
  3. 3.Department of Financial MathematicsFraunhofer ITWMKaiserslauternGermany

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