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. Modern approaches aim at incorporating so-called probabilistic constraints into optimization models for power management and trade. Moreover, the obtained mathematical insight in structure, numerics and stability of probabilistic constraints is applied at the same time to optimization problems of gas networks under uncertainties.

Highlights

The research on optimization problems with probabilistic constraints led to an intensive cooperation with EDF (Electricité de France) and with CEPEL (Rio de Janeiro). The main interest here is in hydro power management under stochastic constraints for reservoir levels or demand satisfaction. Beyond that, appropriate models of probabilistic constraints are supposed to be applied in a fourthcoming DFG-funded research project SFB/Transregio on gas networks under uncertain entry and exit loads. The mathematical focus lies on the structural analysis, modeling and development of algorithms for solving such problems. Particularl challenges here are the transitions from linear to nonlinear and from static to dynamic probabilistic constraints. Applications to renewable energies have shown that such complicated inequality constraints can be dealt with in nontrivial dimension. On the theoretical side, major work is devoted to the derivation of dual stationarity conditions for equilibria in electricity spot markets.

Publications

  Monographs

  • H. Heitsch, R. Henrion, H. Leövey, R. Mirkov, A. Möller, W. Römisch, I. Wegner-Specht, Chapter 13: Empirical Observations and Statistical Analysis of Gas Demand Data, in: Evaluating Gas Network Capacities, Th. Koch, B. Hiller, M.E. Pfetsch, L. Schewe, eds., MOS-SIAM Series on Optimization, SIAM, Philadelphia, 2015, pp. 273--290, (Chapter Published).

  • B. Hiller, Ch. Hayn, H. Heitsch, R. Henrion, H. Leövey, A. Möller, W. Römisch, Chapter 14: Methods for Verifying Booked Capacities, in: Evaluating Gas Network Capacities, Th. Koch, B. Hiller, M.E. Pfetsch, L. Schewe, eds., MOS-SIAM Series on Optimization, SIAM, Philadelphia, 2015, pp. 291--315, (Chapter Published).

  Articles in Refereed Journals

  • H. Heitsch, H. Leövey, W. Römisch, Are quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?, Computational Optimization and Applications. An International Journal, 65 (2016) pp. 567--603.
    Abstract
    Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction.

  • C. Gotzes, H. Heitsch, R. Henrion, R. Schultz, On the quantification of nomination feasibility in stationary gas networks with random load, Mathematical Methods of Operations Research, 84 (2016) pp. 427--457.
    Abstract
    The paper considers the computation of the probability of feasible load constellations in a stationary gas network with uncertain demand. More precisely, a network with a single entry and several exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of an existing flow meeting these loads along with given pressure bounds in the pipes. In a first step, feasibility of deterministic exit loads is characterized algebraically and these general conditions are specified to networks involving at most one cycle. This prerequisite is essential for determining probabilities in a stochastic setting when exit loads are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand. The key of our approach is the application of the spheric-radial decomposition of Gaussian random vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical results are illustrated for different network examples and demonstrate a clear superiority in terms of precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values even for moderate sample size.

  • M. Hintermüller, Th. Surowiec, A bundle-free implicit programming approach for a class of elliptic MPECs in function space, Mathematical Programming Series A, 160 (2016) pp. 271--305.

  • I. Bremer, R. Henrion, A. Möller, Probabilistic constraints via SQP solver: Application to a renewable energy management problem, Computational Management Science, 12 (2015) pp. 435--459.
    Abstract
    The aim of this paper is to illustrate the efficient solution of nonlinear optimization problems with joint probabilistic constraints by means of an SQP method. Here, the random vector is assumed to obey some multivariate Gaussian distribution. The numerical solution approach is applied to a renewable energy management problem. We consider a coupled system of hydro and wind power production used in order to satisfy some local demand of energy and to sell/buy excessive or missing energy on a day-ahead and intraday market, respectively. A short term planning horizon of 2 days is considered and only wind power is assumed to be random. In the first part of the paper, we develop an appropriate optimization problem involving a probabilistic constraint reflecting demand satisfaction. Major attention will be payed to formulate this probabilistic constraint not directly in terms of random wind energy produced but rather in terms of random wind speed, in order to benefit from a large data base for identifying an appropriate distribution of the random parameter. The second part presents some details on integrating Genz' code for Gaussian probabilities of rectangles into the environment of the SQP solver SNOPT. The procedure is validated by means of a simplified optimization problem which by its convex structure allows to estimate the gap between the numerical and theoretical optimal values, respectively. In the last part, numerical results are presented and discussed for the original (nonconvex) optimization problem.

  • A. Fügenschuh, B. Geissler, Ch. Hayn, R. Henrion, B. Hiller, J. Humpola, Th. Koch ET AL., Mathematical optimization for challenging network planning problems in unbundled liberalized gas markets, Energy Systems, 5 (2014) pp. 449--473.

  • W. VAN Ackooij, R. Zorgati, R. Henrion, A. Möller, Joint chance constrained programming for hydro reservoir management, Optimization and Engineering. International Multidisciplinary Journal to Promote Optimization Theory & Applications in Engineering Sciences, 15 (2014) pp. 509--531.

  • L. Andrieu, R. Henrion, W. Römisch, A model for dynamic chance constraints in hydro power reservoir management, European Journal of Operational Research, 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, Mathematical Methods of Operations Research, 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, Applications of Mathematics, 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

  • TH. Arnold, R. Henrion, M. Grötschel, W. Römisch ET AL., B4 -- A Jack of all trades? Solving stochastic mixed-integer nonlinear constraint programs, in: MATHEON -- Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 135--146.

  • 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

  • M. Eigel, J. Neumann, R. Schneider, S. Wolf, Stochastic topology optimisation with hierarchical tensor reconstruction, Preprint no. 2362, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2362 .
    Abstract, PDF (8552 kByte)
    A novel approach for risk-averse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach.

  • M. Hintermüller, C.N. Rautenberg, M. Mohammadi, M. Kanitsar, Optimal sensor placement: A robust approach, Preprint no. 2287, WIAS, Berlin, 2016.
    Abstract, PDF (4835 kByte)
    We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests.

  • H. Egger, Th. Kugler, N. Strogies, Parameter identification in a semilinear hyperbolic system, Preprint no. 2278, WIAS, Berlin, 2016.
    Abstract, PDF (424 kByte)
    We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and contraction arguments. We then investigate the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a variational regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized solutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings.

  Talks, Poster

  • H. Heitsch, Nonlinear probabilistic constraints in gas transportation problems, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, Australia, May 11, 2016.

  • H. Heitsch, Optimization in gas transport networks using nonlinear probabilistic constraints, XIV International Conference on Stochastic Programming (ICSP 2016), Thematic Session: Probabilistic Constraints: Applications and Theory, June 25 - July 1, 2016, Búzios, Brazil, June 28, 2016.

  • J. Neumann, The phase field approach for topology optimization under uncertainties, ZIB Computational Medicine and Numerical Mathematics Seminar, Konrad-Zuse-Zentrum für Informationstechnik Berlin, August 25, 2016.

  • I. Bremer, Dealing with probabilistic constraints under multivariate normal distribution in a standard SQP solver by using Genz' method, WIAS-PGMO Workshop on Nonsmooth and Stochastic Optimization with Applications to Energy Management, May 10 - 12, 2016, WIAS Berlin, May 11, 2016.

  • R. Henrion, Formules du gradient pour des fonctions probabilistes Gaussiennes, Workshop on Offshore Wind Generation, September 9, 2016, Electricité de France R&D, Paris, France, September 9, 2016.

  • R. Henrion, Optimisation sous contraintes en probabilité, Séminaire du Groupe de Travail ``Modèles Stochastiques en Finance'', Ecole Nationale Supérieure des Techniques Avancées (ENSTA) ParisTech, Palaiseau, France, November 28, 2016.

  • R. Henrion, Robust-stochastic optimization problems in stationary gas networks, Conference ``Mathematics of Gas Transport'', October 6 - 7, 2016, Zuse Institut Berlin, October 6, 2016.

  • M. Hintermüller, S. Hajian, N. Strogies, Subproject B02 -- Parameter id., sensor localization and quantification of uncertainties in switched PDE systems, Annual Meeting of the Collaborative Research Center/Transregio (TRR) 154 ``Mathematical Modeling, Simulation and Optimization Using the Example of Gas Networks'', Technische Universität Berlin, October 4 - 5, 2016.

  • M. Hintermüller, S. Hajian, N. Strogies, Subproject B02 -- Parameter id., sensor localization and quantification of uncertainties in switched PDE systems, Conference ``Mathematics of Gas Transport'', Konrad-Zuse-Zentrum für Informationstechnik Berlin, October 6 - 7, 2016.

  • M. Hintermüller, Optimal control of multiphase fluids and droplets, The Fifth International Conference on Continuous Optimization, Session: ``Recent Developments in PDE-constrained Optimization I'', August 6 - 11, 2016, Tokyo, Japan, August 10, 2016.

  • M. Hintermüller, Recent trends in optimal control problems with nonsmooth structures, Computational Methods for Control of Infinite-dimensional Systems, March 14 - 18, 2016, Institute for Mathematics and its Applications, Minneapolis, USA, March 14, 2016.

  • M. Hintermüller, Towards sharp stationarity conditions for classes of optimal control problems for variational inequalities of the second kind, International INdAM Conference ``Optimal Control for Evolutionary PDEs and Related Topics (OCERTO 2016)'', June 20 - 24, 2016, Cortona, Italy, June 20, 2016.

  • H. Heitsch, Optimization of booked capacity in gas transport networks using nonlinear probabilistic constraints, 2nd International Symposium on Mathematical Programming (ISMP 2015), Cluster ``Optimization in Energy Systems'', July 13 - 17, 2015, Pittsburgh, USA, July 17, 2015.

  • R. Henrion, (Sub-)Gradient formulae for probability functions with applications to power management, Universidad de Chile, Centro de Modelamiento Matemático, Santiago de Chile, Chile, November 25, 2015.

  • R. Henrion, Application of chance constraints in a coupled model of hydro-wind energy production, Charles University in Prague, Faculty of Mathematics and Physics, Czech Republic, March 6, 2014.

  • R. Henrion, Application of probabilistic constraints to problems of energy management under uncertainty, Eidgenössische Technische Hochschule Zürich, Power Systems Laboratory, Switzerland, September 30, 2014.

  • R. Henrion, Nonlinear programming for solving chance constrained optimization problems: Application to renewable energies, Winter School on Stochastic Programming with Applications in Energy, Finance and Insurance, March 23 - 28, 2014, Bad Hofgastein, Austria, March 25, 2014.

  • R. Henrion, Probabilistic constraints in hydro reservoir management, XIII Symposium of Specialists in Electric Operational and Expansion Planning (SEPOPE), May 18 - 21, 2014, Foz do Iguassu, Brazil, May 19, 2014.

  • R. Henrion, Probabilistic constraints via nonlinear programming: Application to energy management problems, Euro Mini Conference on Stochastic Programming and Energy Applications (EuroCSP2014), September 24 - 26, 2014, Institut Henri Poincaré, Paris, France, September 25, 2014.

  • 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.