Regularized Decomposition of High-Dimensional Multistage Stochastic Programs with Markov Uncertainty

We develop a quadratic regularization approach for the solution of high-dimensional multistage stochastic optimization problems characterized by a potentially large number of time periods/stages (e.g. hundreds), a high-dimensional resource state variable, and a Markov information process. The resulting algorithms are shown to converge to an optimal policy after a finite number of iterations under mild technical assumptions. Computational experiments are conducted using the setting of optimizing energy storage over a large transmission grid, which motivates both the spatial and temporal dimensions of our problem. Our numerical results indicate that the proposed methods exhibit significantly faster convergence than their classical counterparts, with greater gains observed for higher-dimensional problems

Risk-Averse Model Predictive Operation Control of Islanded Microgrids

In this paper we present a risk-averse model predictive control (MPC) scheme for the operation of islanded microgrids with very high share of renewable energy sources. The proposed scheme mitigates the effect of errors in the determination of the probability distribution of renewable infeed and load. This allows to use less complex and less accurate forecasting methods and to formulate low-dimensional scenario-based optimisation problems which are suitable for control applications. Additionally, the designer may trade performance for safety by interpolating between the conventional stochastic and worst-case MPC formulations. The presented risk-averse MPC problem is formulated as a mixed-integer quadratically-constrained quadratic problem and its favourable characteristics are demonstrated in a case study. This includes a sensitivity analysis that illustrates the robustness to load and renewable power prediction errors

Time-consistent approximations of risk-averse multistage stochastic optimization problems

In this work we study the concept of time consistency as it relates to multistage risk-averse stochastic optimization problems on finite scenario trees. We use dynamic time-consistent formulations to approximate problems having a single global coherent risk measure applied to the aggregated costs over all time periods. The duality of coherent risk measures is employed to create a time-consistent cutting plane algorithm for the construction of non-parametric time-consistent approximations where every one- step conditional risk measure is specified only by its dual representation. Moreover, we show that the method can be extended to generate parametric approximations involving compositions of risk measures from a specified family. Additionally, we also consider the case when the objective function is the mean-upper semideviation measure of risk and develop methods for the construction of universal time-consistent upper bounding functions. We prove that such functions provide time-consistent upper bounds to the global risk measure for an arbitrary feasible policy. Finally, the quality of the approximations generated by the proposed methods is analyzed in multiple computational experiments involving two-stage scenario trees with both artificial data, as well as stock return data for the components of the Dow Jones Industrial Average stock market index. Our numerical results indicate that the dynamic time-consistent formulations closely approximate the original problem for a wide range of risk aversion parameters.Ph. D.Includes bibliographical referencesIncludes vitaby Tsvetan Asamo

Numerical methods in stochastic and two-scale shape optimization

In this thesis, three different models of elastic shape optimization are described. All models use phase fields to describe the elastic shapes and regularize the interface length on some level or scale to control fine-scale structures. First, the paradigm of stochastic dominance is transferred from finite dimensional stochastic programming to elastic shape optimization under stochastic loads. The shapes are optimized under the constraint, that they dominate a given benchmark shape in a certain stochastic order. This allows for a flexible risk aversion comparison. Risk aversion is handled in the constraint rather than the objective functional, which results in an optimization over a subset of admissible shapes only. First and second order stochastic dominance constraints are examined and compared. An (adaptive) Q1 finite element scheme is used, that was implemented for two of the models described in this thesis and is introduced here. Several stochastic loads setups and benchmark variables are discretized and optimized. Starting with the observation that unregularized elastic shape optimization methods create arbitrarily fine micro-structures in many scenarios, domains composited of a number of geometrical subdomains with prescribed boundary conditions are considered in the second model. A reference subdomain is mapped to each type of geometrical subdomains to optimize computational complexity. These are suitable to model fine-scale elastic structures, that are widespread in nature. Examples are fine-scale structures in bones or plants, resulting from the need for a stiff and low-weight structure. The subdomains are coupled to simulate fine-scale structures as they appear e. g. in bones (branching periodic structures). The elastic shape is optimized only for those reference subdomains, simulating periodically repeating structures in one or more coordinate directions. The stress is supposed to be continuous over the domain. A stress-based finite volume discretization and an alternating optimization algorithm are used to find optimal elastic structures for compression and shear loads. Finally, a model considering a fine-scale material in which the elastic shape is modeled by a phase field on the microscale is introduced. This approach further investigates the fine-scale structures mentioned above and allows for a comparison with laminated materials and previous work on homogenization. A short introduction into homogenization is given and the two-scale energies required for the optimization are derived and discretized. An estimation of the scale between macro- and microscale is derived and a finite element discretization using the Heterogeneous Multiscale Method is introduced. Numerical results for compression and shear loads are presented