Application "Optimization Problems in Energy Economics"

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The management of electricity portfolios includes the unit commitment problem (with respect to a network of thermal, hydro, decentralized and other power units) but also trading of electrical energy under the conditions of the liberalized power market. Though already the underlying deterministic problem is not a simple one (including continuous and discrete decision variables), the main challenge here comes from substantial uncertainty of several of its basic data (demands, prices for buying and selling, precipitation etc.). This leads to the consideration of various types of stochastic programming problems. 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 (e.g., risk measures or probabilistic constraints) into stochastic optimization models. This topic is investigated together with the HU Berlin in a joint project, funded by the DFG Research Center MATHEON. Here, the focus of the WIAS research is on optimization of hydro power management under probabilistic constraints and on the analysis of equilibria in oligopolistic power markets (e.g. ISO, EEX).


Highlights

The research on optimization problems with probabilistic constraints led to an intensive cooperation with EDF (Electricité de France) in the framework of a consultation contract. The main interest here is on hydro power management under stochastic constraints for reservoir levels or demand satisfaction. Going beyond the still preferred in water management but disputable approach of so-called individual probabilistic constraints, it could be shown that the appropriate yet more challenging model of joint probabilistic constraints can be successfully treated in meaningful dimensions. Another important progress was made by extending the conventional static model of probabilistic constraints to a dynamic one. In the context of mixed-integer two-stage stochastic programming - which is a relevant mathematical framework for unit commitment problems in power production - a breakthrough in stability oriented reduction of scenario trees could be achieved by basing optimal selection and reweighting of scenarios on so-called polyhedral discrepancy metrics.

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Contributing Groups of WIAS

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Mathematical Context

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Projects/Grants

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Related main application areas

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Publications

 Articles in Refereed Journals

  • L. Andrieu, R. Henrion, W. Römisch, A model for dynamic chance constraints in hydro power reservoir management, European J. Oper. Res., 207 (2010) pp. 579--589.

  • W. VAN Ackooij, R. Henrion, A. Möller, R. Zorgati, On probabilistic constraints induced by rectangular sets and multivariate normal distributions, Math. Methods Oper. Res., 71 (2010) pp. 535--549.

  • R. Henrion, W. Römisch, On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling, Appl. Math., 522 (2007) pp. 473--494.
    Abstract

    Modeling several competitive leaders and followers acting in an electricity market leads to coupled systems of mathematical programs with equilibrium constraints, called equilibrium problems with equilibrium constraints (EPECs). We consider a simplified model for competition in electricity markets under uncertainty of demand in an electricity network as a (stochastic) multi-leader-follower game. First order necessary conditions are developed for the corresponding stochastic EPEC based on a result of Outrata [17]. For applying the general result an explicit representation of the co-derivative of the normal cone mapping to a polyhedron is derived (Proposition 3.2). Later the co-derivative formula is used for verifying constraint qualifications and for identifying M-stationary solutions of the stochastic EPEC if the demand is represented by a finite number of scenarios.

 Contributions to Collected Editions

  • H. Heitsch, R. Henrion, Ch. Küchler, W. Römisch, Generierung von Szenariobäumen und Szenarioreduktion für stochastische Optimierungsprobleme in der Energiewirtschaft, in: Dezentrale regenerative Energieversorgung: Innovative Modellierung und Optimierung, R. Schultz, H.-J. Wagner, eds., LIT Verlag, Münster, 2009, pp. 227--254.

 Preprints, Reports, Technical Reports

  • K. Emich, R. Henrion, W. Römisch, Conditioning of linear-quadratic two-stage stochastic optimization problems, Preprint no. 1783, WIAS, Berlin, 2013.
    Abstract, PDF (194 kByte)

    In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.

 Talks, Poster

  • A. Möller, Probabilistic programming in hydro power management, International Conference Operations Research ``Mastering Complexity'', September 1 - 3, 2010, München, September 1, 2010.

  • R. Henrion, A model for dynamic chance constraints in water reservoir management, 23rd European Conference on Operational Research (EURO23), July 6 - 8, 2009, Bonn, July 7, 2009.

  • R. Henrion, Characterization of M-stationary points for an equilibrium problem in an electricity spot market model, 16th International Conference on Mathematical Methods in Economics and Industry, June 15 - 18, 2009, České Budějovice, Czech Republic, June 17, 2009.

  • R. Henrion, On stationarity conditions for an equilibrium problem with equilibrium constraints from an electricity spot market model, 23rd European Conference on Operational Research (EURO23), July 6 - 8, 2009, Bonn, July 7, 2009.

  • R. Henrion, On a dynamical model for chance constrained programming, Conference on Optimization & Practices in Industry (COPI08), November 26 - 28, 2008, Clamart, France, November 28, 2008.

  • R. Henrion, Contraintes en probabilité: synthèse bibliographique et approche à la situation dynamique, Electricité de France R&D, Clamart, France, November 28, 2007.

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