Summary
Dynamic stochastic optimization techniques are highly relevant for applications in electricity production and trading since there are uncertainty factors at different time stages (e.g., demand, spot prices) that can be described reasonably by statistical models. In this paper, two aspects of this approach are highlighted: scenario tree approximation and risk aversion. The former is a procedure to replace a general statistical model (probability distribution), which makes the optimization problem intractable, suitably by a finite discrete distribution. Our methods rest upon suitable stability results for stochastic optimization problems. With regard to risk aversion we present the approach of polyhedral risk measures. For stochastic optimization problems minimizing risk measures from this class it has been shown that numerical tractability as well as stability results known for classical (nonrisk-averse) stochastic programs remain valid. In particular, the same scenario approximation methods can be used. Finally, we present illustrative numerical results from an electricity portfolio optimization model for a municipal power utility.
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References
P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk.Mathematical Finance, 9:203–228, 1999.
P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, and H. Ku. Coherent multiperiod risk adjusted values and Bellman's principle.Annals of Operations Research,152:5–22, 2007.
B. Blaesig.Risikomanagement in der Stromerzeugungs—und Handelsplanung, volume 113 ofAachener Beitrge zur Energieversorgung. Klinkenberg, Aachen, Germany, 2007. PhD Thesis.
M.S. Casey and S. Sen. The scenario generation algorithm for multistage stochastic linear programming.Mathematics of Operations Research, 30:615– 631, 2005.
M.A.H. Dempster. Sequential importance sampling algorithms for dynamic stochastic programming.Zapiski Nauchnykh Seminarov POMI, 312:94–129, 2004.
J. Dupačová, G. Consigli, and S.W. Wallace. Scenarios for multistage stochastic programs.Annals of Operations Research, 100:25–53, 2000.
A. Eichhorn.Stochastic Programming Recourse Models: Approximation, Risk aversion, Applications in Energy. PhD thesis, Department of Mathematics, Humboldt University, Berlin, 2007.
A. Eichhorn and W. Römisch. Polyhedral risk measures in stochastic programming.SIAM Journal on Optimization, 16:69–95, 2005.
A. Eichhorn and W. Römisch. Mean-risk optimization models for electricity portfolio management. InProceedings of the 9th International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Stockholm, Sweden, 2006.
A. Eichhorn and W. Römisch. Stability of multistage stochastic programs incorporating polyhedral risk measures.Optimization, 57:295–318, 2008.
A. Eichhorn, W. Römisch, and I. Wegner. Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. InIEEE St. Petersburg PowerTech Proceedings, 2005.
S.-E. Fleten and T.K. Kristoffersen. Short-term hydropower production planning by stochastic programming.Computers and Operations Research, 35:2656– 2671, 2008.
S.-E. Fleten, S.W. Wallace, and W.T. Ziemba. Hedging electricity portfolios via stochastic programming. In C. Greengard and A. Ruszczyński, editors,Decision Making under Uncertainty: Energy and Power, volume 128 ofIMA Volumes in Mathematics and its Applications. Springer, New York, pages 71–93, 2002.
H. Föllmer and A. Schied.Stochastic Finance. An Introduction in Discrete Time, volume 27 ofDe Gruyter Studies in Mathematics. Walter de Gruyter, Berlin, 2nd edition, 2004.
M. Frittelli and G. Scandolo. Risk measures and capital requirements for processes.Mathematical Finance, 16:589–612, 2005.
S. Graf and H. Luschgy.Foundations of Quantization for Probability Distributions, volume 1730 ofLecture Notes in Mathematics. Springer, Berlin, 2000.
C. Greengard and A. Ruszczyński, editors,Decision Making under Uncertainty: Energy and Power, volume 128 ofIMA Volumes in Mathematics and its Applications. Springer, New York, 2002.
N. Gröwe-Kuska, K.C. Kiwiel, M.P. Nowak, W. Römisch, and I. Wegner. Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation. In C. Greengard and A. Ruszczyński, editors,Decision Making under Uncertainty: Energy and Power, volume 128 ofIMA Volumes in Mathematics and its Applications. Springer, New York, pages 39–70, 2002.
N. Gröwe-Kuska and W. Römisch. Stochastic unit commitment in hydro-thermal power production planning. In S. W. Wallace and W. T. Ziemba, editors,Applications of Stochastic Programming, MPS/SIAM Series on Optimization, pages 633–653. SIAM, Philadelphia, PA, USA, 2005.
H. Heitsch.Stabilität und Approximation stochastischer Optimierungsprobleme. PhD thesis, Department of Mathematics, Humboldt University, Berlin, 2007.
H. Heitsch and W. Römisch. Scenario reduction algorithms in stochastic programming.Computational Optimization and Applications, 24:187–206, 2003.
H. Heitsch and W. Römisch. Stability and scenario trees for multistage stochastic programs. In G. Infanger, editor,Stochastic Programming—The State of the Art. 2009. to appear.
H. Heitsch and W. Römisch. Scenario tree modeling for multistage stochastic programs.Mathematical Programming, to appear, 2009.
H. Heitsch, W. Römisch, and C. Strugarek. Stability of multistage stochastic programs.SIAM Journal on Optimization, 17:511–525, 2006.
R. Hochreiter and G. Ch. Pflug. Financial scenario generation for stochastic multi-stage decision processes as facility location problems.Annals of Operations Research, 152:257–272, 2007.
R. Hochreiter, G. Ch. Pflug, and D. Wozabal. Multi-stage stochastic electricity portfolio optimization in liberalized energy markets. InSystem Modeling and Optimization, IFIP International Federation for Information Processing. Springer, Boston, MA, USA, pages 219–226, 2006.
K. Høyland, M. Kaut, and S.W. Wallace. A heuristic for moment-matching scenario generation.Computational Optimization and Applications, 24:169–185, 2003.
B. Krasenbrink.Integrierte Jahresplanung von Stromerzeugung und—handel, volume 81 ofAachener Beiträge zur Energieversorgung. Klinkenberg, Aachen, Germany, 2002. PhD Thesis.
W. Ogryczak and A. Ruszczyński. On consistency of stochastic dominance and mean-semideviation models.Mathematical Programming, 89:217–232, 2001.
T. Pennanen. Epi-convergent discretizations of multistage stochastic programs via integration quadratures.Mathematical Programming, 116:461–479, 2008.
M.V.F. Pereira and L.M.V.G. Pinto. Multi-stage stochastic optimization applied to energy planning.Mathematical Programming, 52:359–375, 1991.
G. Ch. Pflug and W. Römisch.Modeling, Measuring, and Managing Risk. World Scientific, Singapore, 2007.
R.T. Rockafellar and S. Uryasev. Conditional value-at-risk for general loss distributions.Journal of Banking and Finance, 26:1443–1471, 2002.
R.T. Rockafellar and R.J-B. Wets.Variational Analysis, volume 317 ofGrundlehren der mathematischen Wissenschaften. Springer, Berlin, 1st edition, 1998. (Corr. 2nd printing 2004).
A. Ruszczyński and A. Shapiro, editors,Stochastic Programming, volume 10 ofHandbooks in Operations Research and Management Science. Elsevier, Amsterdam, 1st edition, 2003.
H.K. Schmöller.Modellierung von Unsicherheiten bei der mittelfristigen Strom-erzeugungs—und Handelsplanung, volume 103 ofAachener Beitrge zur Energie-versorgung. Klinkenberg, Aachen, Germany, 2005. PhD Thesis.
S. Sen, L. Yu, and T. Genc. A stochastic programming approach to power portfolio optimization.Operations Research, 54:55–72, 2006.
S. Takriti, B. Krasenbrink, and L.S.-Y. Wu. Incorporating fuel constraints and electricity spot prices into the stochastic unit commitment problem.Operations Research, 48:268–280, 2000.
J. von Neumann and O. Morgenstern.Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ, USA, 1944.
S.W. Wallace and S.-E. Fleten. Stochastic programming models in energy. In Ruszczyński and Shapiro [35], Chap. 10, vol. 10 of Handbooks in Operations Research and Management Science. North-Holland, The Netherlands, pp. 637– 677, 2003.
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Eichhorn, A., Heitsch, H., Römisch, W. (2009). Scenario Tree Approximation and Risk Aversion Strategies for Stochastic Optimization of Electricity Production and Trading. In: Kallrath, J., Pardalos, P.M., Rebennack, S., Scheidt, M. (eds) Optimization in the Energy Industry. Energy Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88965-6_14
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