Michael Klöppel

Advances and applications of chance-constrained approaches to systems optimisation under uncertainty

A chance-constrained optimisation (CCOPT) model has a dual goal: guaranteeing performance as well as system reliability under uncertainty. The beginning of CCOPT methods dates back in the 1950s. Recently, CCOPT approaches are gaining momentum as modern engineering and finance applications are forced to consider reliability and risk metrics at the design and planning stages. Although theoretical development and practical applications have been made, many open problems remain to be addressed in this area. This article attempts to provide a brief survey of major application areas, structure properties, challenges and solution approaches to CCOPT. In particular, we present our research results achieved in the past few years.

Monotony analysis and sparse-grid integration for nonlinear chance constrained process optimization

The numerical solution of a nonlinear chance constrained optimization problem poses a major challenge. The idea of back-mapping as introduced by M. Wendt, P. Li and G. Wozny in 2002 is a viable approach for transforming chance constraints on output variables (of unknown distribution) into chance constraints on uncertain input variables (of known distribution) based on a monotony relation. Once transformation of chance constraints has been accomplished, the resulting optimization problem can be solved by using a gradient-based algorithm. However, the computation of values and gradients of chance constraints and the objective function involves the evaluation of multi-dimensional integrals, which is computationally very expensive. This study proposes an easy-to-use method for analysing monotonic relations between constrained outputs and uncertain inputs. In addition, sparse-grid integration techniques are used to reduce the computational time decisively. Two examples from process optimization under uncertainty demonstrate the performance of the proposed approach.

A tractable approximation of non-convex chance constrained optimization with non-Gaussian uncertainties

Chance constrained optimization problems in engineering applications possess highly nonlinear process models and non-convex structures. As a result, solving a nonlinear non-convex chance constrained optimization (CCOPT) problem remains as a challenging task. The major difficulty lies in the evaluation of probability values and gradients of inequality constraints which are nonlinear functions of stochastic variables. This article proposes a novel analytic approximation to improve the tractability of smooth non-convex chance constraints. The approximation uses a smooth parametric function to define a sequence of smooth nonlinear programs (NLPs). The sequence of optimal solutions of these NLPs remains always feasible and converges to the solution set of the CCOPT problem. Furthermore, Karush–Kuhn–Tucker (KKT) points of the approximating problems converge to a subset of KKT points of the CCOPT problem. Another feature of this approach is that it can handle uncertainties with both Gaussian and/or non-Gaussian distributions.

Chance constrained optimal power flow with non-Gaussian distributed uncertain wind power generation

Chance constrained optimization (CCOPT) turns out to be a valuable approach to optimal power flow under uncertainty. The aim of this study is to develop a novel computation framework to solve CCOPT problems with non-Gaussian distributed uncertain variables and apply it to optimal power flow (OPF) under uncertain wind power penetration with Beta distribution. Results of OPF of a real distribution system with embedded wind power generation show the effectiveness of the proposed approach.

Dynamics of the laser-induced nanostructuring of thin metal layers: experiment and theory

Nanostructures are of increasing importance in manifold application fields such as electronics, optics and beyond. However, the fast and cost-effective production of nanostructures is a big technological challenge for laser machining. One promising approach is laser irradiation of thin metal layers, which allows the fabrication of metal nanostructures induced by a melting and transformation process. The influence of laser parameters (laser fluence, laser pulse number) on the morphology of the nanopatterned film and the dynamics of the nanostructure formation during excimer laser irradiation of a 20 nm chromium film on fused silica were studied. The dynamics of nanopatterning, comprising hole and droplet formation, were investigated by time-dependent reflection and transmission measurements as well as time-dependent optical microscopy. The resulting patterns were investigated by optical and scanning electron microscopy (SEM). However, for an optimization of this process a better understanding of the underlying physical phenomena is necessary. Therefore, experimental data of laser-induced nanopatterning were compared with results of physical simulations that consider the heat equation (laser–solid interaction including melting and evaporation) and the Navier–Stokes equation (transformation processes of the molten phase). The simulations, making use of laser fluence-dependent effective material parameters (surface tension and viscosity), are in good agreement with the experimental results.

Evaluating integration approaches to robust process optimization and control using chance constrained programming

This contribution concerns robust optimization and control of nonlinear steady-state and dynamic processes under uncertain disturbances or parameters. We use chance constrained programming to solve such stochastic optimization and control problems. While steady-state processes always possess time-independent uncertain parameters, in dynamic processes there may be both time-independent and time-dependent uncertain parameters. For problems with time-dependent uncertain parameters these parameters will be discretized in the time horizon and thus the total number of uncertain variables to be treated will be high, which leads to difficulties to compute probabilities and their gradients. A new approach to an efficient computation is developed by using the sparse grid technique with which CPU-time can be significantly reduced. Two application examples from process engineering are taken to demonstrate the effectiveness of our computational framework. The performance of sparse grid integration is verified through numerical experiments in comparison to Monte-Carlo and Quasi-Monte-Carlo techniques.

An Inner-Outer Approximation Approach to Chance Constrained Optimization

Nonlinear chance constrained optimization (CCOPT) problems are known to be difficult to solve. This work proposes a smooth approximation approach consisting of an inner and an outer analytic approximation of chance constraints. In this way, CCOPT is approximated by two parametric nonlinear programming (NLP) problems which can be readily solved by an NLP solver. Any optimal solution of the inner approximation problem is a priori feasible to the CCOPT. The solutions of the inner and outer problems, respectively, converge asymptotically to the optimal solution of the CCOPT.