Optimal control of electricity input given an uncertain demand

Abstract

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.

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Notes

  1. 1.

    https://www.next-kraftwerke.com/knowledge/intraday-trading, last checked: 2nd of April, 2019.

  2. 2.

    https://de.mathworks.com/help/optim/ug/fmincon.html, last checked: Sept 21, 2018.

References

  1. Aïd R, Campi L, Huu AN, Touzi N (2009) A structural risk-neutral model of electricity prices. Int J Theor Appl Financ 12:925–947

    MathSciNet  Article  Google Scholar 

  2. Annunziato M, Borzì A (2013) A Fokker–Planck control framework for multidimensional stochastic processes. J Comput Appl Math 237:487–507

    MathSciNet  Article  Google Scholar 

  3. Annunziato M, Borzì A (2018) A Fokker–Planck control framework for stochastic systems. EMS Surv Math Sci 5:65–98

    MathSciNet  Article  Google Scholar 

  4. Applebaum D (2009) Lévy processes and stochastic calculus, vol. 116 of Cambridge studies in advanced mathematics, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  5. Barlow MT (2002) A diffusion model for electricity prices. Math Financ 12:287–298

    Article  Google 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., Hackensack

    Google 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 20:435–462

    MathSciNet  Article  Google 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 37:980–995

    MathSciNet  Article  Google Scholar 

  10. Göttlich S, Teuber C (2018) Space mapping techniques for the optimal inflow control of transmission lines. Optim Methods Softw 33:120–139

    MathSciNet  Article  Google Scholar 

  11. Kiesel R, Schindlmayr G, Börger RH (2009) A two-factor model for the electricity forward market. Quant Financ 9:279–287

    MathSciNet  Article  Google Scholar 

  12. Klenke A (2008) Probability theory: a comprehensive course. Springer, London

    Google 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 Raton

    Google Scholar 

  14. La Marca M, Armbruster D, Herty M, Ringhofer C (2010) Control of continuum models of production systems. IEEE Trans Automat Control 55:2511–2526

    MathSciNet  Article  Google Scholar 

  15. LeVeque RJ (1990) Numerical methods for conservation laws, lectures in mathematics ETH Zürich. Birkhäuser Verlag, Basel

    Google Scholar 

  16. Lucia JJ, Schwartz ES (2002) Electricity prices and power derivatives: evidence from the Nordic power exchange. Rev Deriv Res 5:5–50

    Article  Google Scholar 

  17. Mikosch T (2009) Non-life insurance mathematics: an introduction with the Poisson process, 2nd edn. Universitext, Springer, Berlin

    Google Scholar 

  18. Roy S, Annunziato M, Borzì A, Klingenberg C (2018) A Fokker–Planck approach to control collective motion. Comput Optim Appl 69:423–459

    MathSciNet  Article  Google Scholar 

  19. Schwartz E, Smith JE (2000) Short-term variations and long-term dynamics in commodity prices. Manag Sci 46:893–911

    Article  Google Scholar 

  20. Wagner A (2014) Residual demand modeling and application to electricity pricing. Energy J 35:45–73

    Article  Google Scholar 

Download references

Acknowledgements

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.

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6 Appendix

6 Appendix

The detailed calculation of the second moment of the JDP used in Sect. 4.1 is as follows:

$$\begin{aligned} {\mathbb {E}}\left[ Y_t^2\right]&= {\mathbb {E}}\left[ \left( e^{-\kappa t}y_0 + \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds + \sum _{i=1}^{N_t} \gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right] \nonumber \\&= {{\mathbb {E}}\left[ \underbrace{\left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) ^2}_{=A} + \underbrace{\left( \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s + \sum _{i=1}^{N_t} \gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2}_{=B}\right. }\nonumber \\&\quad {\left. +\,\underbrace{2\left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) \cdot \left( \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s + \sum _{i=1}^{N_t} \gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) }_{=C} \right] } \end{aligned}$$

For a better readability, we evaluate A, B, and C in expectation separately:

$$\begin{aligned} {\bar{A} = {\mathbb {E}}[A]}&= \left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) ^2\\&=y_0^2e^{-2\kappa t} + 2y_0e^{-\kappa t} \int _{0}^{t}e^{-\kappa (t-s)}\kappa \mu (s) ds + \left( \kappa \int _{0}^{t}e^{-\kappa (t-s)} \mu (s) ds\right) ^2. \end{aligned}$$

For the diffusion- and jump-driven quadratic term B, we calculate

$$\begin{aligned} {\bar{B} = {\mathbb {E}}[B]}&= {\mathbb {E}}\left[ \left( \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s\right) ^2\right] +{\mathbb {E}}\left[ \left( \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right] \\&\quad +\,2{\mathbb {E}}\left[ \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s\sum _{i=1}^{N_t} \gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right] \\&= {\mathbb {E}}\left[ \sigma ^2 \int _{0}^{t} e^{-2 \kappa (t-s)} ds\right] +{\mathbb {E}}\left[ \left( \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right] \\&\quad +\,2{\mathbb {E}}\left[ \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s\right] {\mathbb {E}}\left[ \sum _{i=1}^{N_t} \gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right] \\&= \sigma ^2 \left[ \frac{e^{-2 \kappa (t-s)}}{2\kappa }\right] _{s=0}^{s=t} + {\mathbb {E}}\left[ \left( \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right] \\&= \frac{\sigma ^2}{2\kappa }\left( 1-e^{-2 \kappa t}\right) + {\mathbb {E}}\left[ \left( \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right] . \end{aligned}$$

Note that the expectation of the integral with respect to the Brownian motion is zero. It remains to calculate the expectation of the mixed summand C:

$$\begin{aligned} {\bar{C} = {\mathbb {E}}[C]}&= 2\left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) \cdot {\mathbb {E}}\left[ \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s + \sum _{i=1}^{N_t} \gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right] \\&= 2\left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) \cdot {\mathbb {E}}\left[ \sigma \int _{0}^{t} e^{-\kappa (t-s)} dW_s\right] \\&\quad +\,2\left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) \cdot {\mathbb {E}}\left[ \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right] \\&= 2\left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) \cdot {\mathbb {E}}\left[ \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right] \\&= 2\left( e^{-\kappa t}y_0 + \kappa \int _{0}^{t} e^{-\kappa (t-s)}\mu (s) ds\right) \cdot {\bar{\gamma }} \frac{\nu }{\kappa }\left( 1-e^{-\kappa t}\right) . \end{aligned}$$

Again the expectation of the integral with respect to the Brownian motion vanishes. Note that the second moment of the time-dependent OUP is obtained by setting \(\gamma _{\tau _i} \equiv 0\) for all jump times \(\tau _i\).

Thus, it remains to calculate \({\mathbb {E}}\left[ \left( \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right] \). Note that \({\mathbb {E}}\left[ \gamma \right] ^2=\bar{\gamma }^2\).

$$\begin{aligned} {\mathbb {E}}\left[ \left( \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right]&= {\mathbb {E}}\left[ \sum _{i=1}^{N_t}\gamma _{\tau _i}^2 e^{-2\kappa (t-\tau _i)} + \sum _{i\ne j}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\gamma _{\tau _j} e^{-\kappa (t-\tau _j)}\right] \\&= {\mathbb {E}}\left[ \sum _{i=1}^{N_t} e^{-2\kappa (t-\tau _i)}\right] \bar{\gamma }_2+ {\mathbb {E}}\left[ \sum _{i\ne j}^{N_t} e^{-\kappa (t-\tau _i)}e^{-\kappa (t-\tau _j)}\right] \bar{\gamma }^2\\&= {\mathbb {E}}\left[ \sum _{i=1}^{N_t} e^{-2\kappa (t-\tau _i)}\right] \bar{\gamma }_2+ 2\cdot {\mathbb {E}}\left[ {\mathbb {E}}\left[ \sum _{i=2}^{N_t} \sum _{j=1}^{i-1} e^{-\kappa (t-\tau _i)}e^{-\kappa (t-\tau _j)}|N_t\right] \right] \bar{\gamma }^2. \end{aligned}$$

We know from (Mikosch 2009, Prop. 2. 1. 16) that \(\tau _i \sim {\mathcal {U}}[0,t]\). Thus, we calculate

$$\begin{aligned} {\mathbb {E}}\left[ e^{-2\kappa (t-\tau _i)}\right]&= \int _{0}^{t}e^{-2\kappa (t-s)}\frac{1}{t}ds = \frac{1-e^{-2\kappa t}}{2\kappa t}. \end{aligned}$$

We can then deduce

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^{N_t} e^{-2\kappa (t-\tau _i)}\right]&= \sum _{i=1}^{\infty } i \cdot e^{-\nu t} \frac{(\nu t)^{i}}{i !} \frac{1-e^{-2\kappa t}}{2\kappa t} = \nu \cdot \frac{(1-e^{-2\kappa t})}{2\kappa }. \end{aligned}$$

It remains to compute the mixed-term expectation.

$$\begin{aligned} {\mathbb {E}}\left[ e^{-\kappa (t-\tau _i)}e^{-\kappa (t-\tau _j)}\right| \tau _j<\tau _i, N_t=n]&= \int _{0}^{t}e^{-\kappa (t-s)} \frac{1}{t} \int _{0}^{s} e^{-\kappa (t-u)}\frac{2}{t} du ds \\&= \int _{0}^{t} \frac{2\cdot (e^{-2\kappa (t-s)}-e^{-\kappa (2t-s)})}{\kappa t^2} ds \\&= \frac{1+e^{-2\kappa t}-2e^{-\kappa t}}{\kappa ^2 t^2}. \end{aligned}$$

Thus, we have

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=2}^{N_t} \sum _{j=1}^{i-1} e^{-\kappa (t-\tau _i)}e^{-\kappa (t-\tau _j)}|N_t=n\right]&= \sum _{i=2}^{n} \sum _{j=1}^{i-1} {\mathbb {E}}\left[ e^{-\kappa (t-\tau _i)}e^{-\kappa (t-\tau _j)}|N_t=n\right] \\&= \sum _{i=1}^{n-1} i \cdot {\mathbb {E}}\left[ e^{-\kappa (t-\tau _{i+1})}e^{-\kappa (t-t_1)}|N_t=n\right] \\&= {\mathbb {E}}\left[ e^{-\kappa (t-t_{2})}e^{-\kappa (t-t_1)}|N_t=n\right] \cdot \frac{n \cdot (n-1)}{2} \\&= \frac{1+e^{-2\kappa t}-2e^{-\kappa t}}{\kappa ^2 t^2} \cdot \frac{n \cdot (n-1)}{2}.\\ {\mathbb {E}}\left[ {\mathbb {E}}\left[ \sum _{i=2}^{N_t} \sum _{j=1}^{i-1} e^{-\kappa (t-\tau _i)}e^{-\kappa (t-\tau _j)}|N_t\right] \right]&= {\mathbb {E}}\left[ N_t^2-N_t\right] \cdot \frac{1+e^{-2\kappa t}-2e^{-\kappa t}}{2 \kappa ^2 t^2} \\&= \nu ^2 \cdot \frac{1+e^{-2\kappa t}-2e^{-\kappa t}}{2 \kappa ^2}. \end{aligned}$$

Finally, the closed-form expression is

$$\begin{aligned} {\mathbb {E}}\left[ \left( \sum _{i=1}^{N_t}\gamma _{\tau _i} e^{-\kappa (t-\tau _i)}\right) ^2\right]&= \nu \cdot \frac{(1-e^{-2\kappa t})}{2\kappa } \cdot \bar{\gamma }_2+ \nu ^2 \cdot \frac{1+e^{-2\kappa t}-2e^{-\kappa t}}{\kappa ^2} \cdot \bar{\gamma }^2. \end{aligned}$$

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Göttlich, S., Korn, R. & Lux, K. Optimal control of electricity input given an uncertain demand. Math Meth Oper Res 90, 301–328 (2019). https://doi.org/10.1007/s00186-019-00678-6

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Keywords

  • Stochastic optimal control
  • Jump diffusion processes
  • Transport equation

Mathematics Subject Classification

  • 93E20
  • 60H10
  • 65C20