Application Area C: Production
Project C7

Mean-risk optimization of electricity production in liberalized markets

Duration:	September 2002 - May 2014
Project heads:	Werner Römisch and René Henrion
	Department of Mathematics, Humboldt-University Berlin, 10099 Berlin, Germany
	Tel: +49 (0)30 - 2093 2561 (office) / - 2093 2353 (secretary)
	email: romisch@math.hu-berlin.de
	Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
	Tel: +49 (0)30 - 20372 540
	email: henrion@wias-berlin.de
Responsible:	Konstantin Emich
	Department of Mathematics, Humboldt-University Berlin, 10099 Berlin, Germany
	Tel: +49 (0)30 - 2093 5448 (office) / - 2093 2353 (secretary)
	email: emich@math.hu-berlin.de
Cooperation:	GAMS Software
	EDF Electricité de France
Support:	DFG Research Center Matheon "Mathematics for key technologies"

Description :

Traditional models in stochastic programming and in stochastic power management are based on maximizing expected revenues but do not reflect the risk of decisions. In power utilities, portfolio optimization and risk management are typically considered as separate tasks. Since overcoming this separation promises additional efficiency, proposals came to the fore to incorporate risk functionals (risk measures, cf. [1], [2], [23] and Chapter 4 in [20]) into (stochastic) optimization models (cf., e.g., [22], [23], [24]). This leads to mean-risk stochastic programming models, i.e., models that optimize expected revenues and risk simultaneously. Alternatively, one can incorporate probabilistic constraints ([14], [16]) in order to limit risk. However, both types of risk-averse stochastic optimization models show different structural and stability properties than purely expectation based stochastic programs (cf. [26]), thus new algorithmic approaches are needed (cf. [25]).
We identify favorable properties of risk functionals with regard to stability and computational aspects of mean-risk models. Particular emphasis is laid on dynamic models. Structural properties of the models and their implications to scenario selection and the design of decomposition methods are explored. Methods for generating multivariate scenario trees for mean-risk electricity portfolio management models are developed and applied to construct trees containing electrical load, spot price, and inflow [4]. Extending our earlier work ([12]), we develop solution algorithms for mean-risk models in power engineering based on dual decomposition strategies (cf. [12], Chapter 3 in [27]), new subproblem solvers, and Lagrangian heuristics. These algorithms are implemented and tested on real-life data of power exchanges and the hydro-thermal generation system of the cooperation partners. Furthermore, we analyze stability and structural properties of stochastic programs with equilibrium constraints (cf. [19], [28]). Such constraints arise from incorporating market models of, e.g., power exchanges into the optimization model [18].

The quantitative stability analysis of stochastic programs was surveyed [26] and extended for probabilistically constrained [16] and multistage [13] models. In [16] conditions for Hölder and Lipschitz dependence of solution sets on the underlying probability distribution are derived. In [13] it is observed that a suitable distance of probability distributions as well as a distance for the underlying filtrations are needed for proving stability results in the multistage case. For mixed-integer stochastic programs a central limit theorem for Monte Carlo estimators of their infima is derived [6], scenario reduction is studied [15] and progress is obtained for comparing dual decomposition schemes [3]. The central limit result in [6] is used to justify bootstrapping and subsampling techniques for mixed-integer models. In [3] the duality gaps of scenario, node and geograhical decomposition schemes are compared. The class of polyhedral risk measures is introduced and studied in [5]. Furthermore, conditions for their convexity and coherence are derived, and favourable properties with respect to stability [8] and decomposition are discussed. Multiperiod coherent polyhedral risk measures have been tested in optimization models for electricity portfolios [4], [7], [9], [23].

[1]	P. Artzner, F. Delbaen, J.-M. Eber, D. Heath:
	Coherent measures of risk, Mathematical Finance 9 (1999), 203-228.
[2]	P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, H. Ku:
	Coherent multiperiod risk adjusted values and Bellman's principle, Annals of Operations Research 152 (2007), 5--22.
[3]	D. Dentcheva and W. Römisch:
	Duality gaps in nonconvex stochastic optimization, Mathematical Programming 101 (2004), 515--535.
[4]	A. Eichhorn, H. Heitsch and W. Römisch:
	Stochastic optimization of electricity portfolios: Scenario tree modeling and risk management, Preprint 504, DFG Research Center Matheon "Mathematics for key technologies", 2008 and submitted for publication in Power Systems Handbook, Springer.
[5]	A. Eichhorn and W. Römisch:
	Polyhedral risk measures in stochastic programming, SIAM Journal of Optimization 16 (2005), 69--95.
[6]	A. Eichhorn and W. Römisch:
	Stochastic integer programming: Limit theorems and confidence intervals, Mathematics of Operations Research 32 (2007), 118--135.
[7]	A. Eichhorn and W. Römisch:
	Mean-risk optimization models for electricity portfolio management, Proceedings of PMAPS 2006 (Probabilistic Methods Applied to Power Systems), Stockholm (Sweden), June 2006.
[8]	A. Eichhorn and W. Römisch:
	Stability of multistage stochastic programs incorporating polyhedral risk measures, Optimization 57 (2008), 295--318.
[9]	A. Eichhorn and W. Römisch:
	Dynamic risk management in electricity portfolio optimization via polyhedral rik functionals, Preprint 460, DFG Research Center Matheon "Mathematics for key technologies", 2008 and to appear in the Proceedings of the 2008 PES IEEE General Meeting.
[10]	A. Eichhorn, W. Römisch and I. Wegner:
	Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures, IEEE St. Petersburg Power Tech Proceedings 2005.
[11]	H. Föllmer and A. Schied:
	Stochastic Finance: An Introduction in Discrete Time, 2nd ed., De Gruyter Studies in Mathematics, vol. 27, Walter de Gruyter, Berlin, 2004.
[12]	N. Gröwe-Kuska, K.C. Kiwiel, M.P. Nowak, W. Römisch, I. Wegner:
	Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation, in: Decision Making under Uncertainty: Energy and Power (C. Greengard, A. Ruszczynski, eds.), IMA Volumes in Mathematics and its Applications, vol. 128, Springer, New York, 2002, 39-70.
[13]	H. Heitsch, W. Römisch and C. Strugarek:
	Stability of multistage stochastic programs, SIAM Journal on Optimization 17 (2006), 511--525.
[14]	R. Henrion:
	Qualitative stability of convex programs with probabilistic constraints, in: Lect. Notes in Economics and Mathematical Systems, vol. 481: Optimization, (V.H. Nguyen, J.-J. Strodiot and P. Tossings eds.), Springer, Berlin, 2000, pp. 164-180.
[15]	R. Henrion, C. Küchler and W. Römisch:
	Scenario reduction in stochastic programming with respect to discrepancy distances, Computational Optimization and Applications (to appear).
[16]	R. Henrion and W. Römisch:
	Metric regularity and quantitative stability in stochastic programs with probabilistic constraints, Mathematical Programming 84 (1999), 55-88.
[17]	R. Henrion and W. Römisch:
	Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints, Mathematical Programming 100 (2004), 589--611.
[18]	R. Henrion and W. Römisch:
	On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling, Applications of Mathematics 52 (2007), 473--494.
[19]	M. Kocvara and J.V. Outrata:
	Optimization problems with equilibrium constraints and their numerical solution, Mathematical Programming 101 (2004), 119-149.
[20]	A. Müller and D. Stoyan:
	Comparison Methods for Stochastic Models and Risks, Wiley, Chichester, 2002.
[21]	M.P. Nowak and W. Römisch:
	Stochastic Lagrangian relaxation applied to power scheduling in a hydro-thermal system under uncertainty, Annals of Operations Research 100 (2000), 251-272.
[22]	W. Ogryczak and A. Ruszczynski:
	On consistency of stochastic dominance and mean-semideviation models, Mathematical Programming 89 (2001), 217-232.
[23]	G.Ch. Pflug and W. Römisch:
	Modeling, Measuring and Managing Risk, World Scientific, Singapore, 2007.
[24]	R.T. Rockafellar and S. Uryasev:
	Conditional value-at-risk for general loss distributions, Journal of Banking & Finance 26 (2002), 1443-1471.
[25]	W. Römisch:
	Optimierungsmethoden für die Energiewirtschaft: Stand und Entwicklungstendenzen, in: Optimierung in der Energieversorgung, VDI-Berichte 1627, VDI-Verlag, Düsseldorf, 2001, 23-36.
[26]	W. Römisch:
	Stability of Stochastic Programming Problems, in: Stochastic Programming (A. Ruszczynski and A. Shapiro eds.), Handbooks in Operations Research and Management Science Vol. 10, Elsevier, Amsterdam 2003, 483-554.
[27]	A. Ruszczynski and A. Shapiro (eds.):
	Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10, Elsevier, Amsterdam, 2003.
[28]	A. Shapiro:
	Stochastic programming with equilibrium constraints, Journal of Optimization Theory and Applications 128 (2006), 221--243.

last modified April 4, 2008