## 510 Mathematik

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In this paper, we consider the question of data aggregation using the practical example of emissions data for economic activities for the sustainability assessment of regional bank clients. Given the current scarcity of company-specific emission data, an approximation relies on using available public data. These data are reported in different standards in different sources. To determine a mapping between the different standards, an adaptation to the Covariance Matrix Self-Adaptation Evolution Strategy is proposed. The obtained results show that high-quality mappings are found. Nevertheless, our approach is transferable to other data compatibility problems. These can be found in the merging of emissions data for other countries, or in bridging the gap between completely different data sets.

Alleviating the curse of dimensionality in minkowski sum approximations of storage flexibility
(2023)

Many real-world applications require the joint optimization of a large number of flexible devices over some time horizon. The flexibility of multiple batteries, thermostatically controlled loads, or electric vehicles, e.g., can be used to support grid operations and to reduce operation costs. Using piecewise constant power values, the flexibility of each device over d time periods can be described as a polytopic subset in power space. The aggregated flexibility is given by the Minkowski sum of these polytopes. As the computation of Minkowski sums is in general demanding, several approximations have been proposed in the literature. Yet, their application potential is often objective-dependent and limited by the curse of dimensionality. In this paper, we show that up to 2d vertices of each polytope can be computed efficiently and that the convex hull of their sums provides a computationally efficient inner approximation of the Minkowski sum. Via an extensive simulation study, we illustrate that our approach outperforms ten state-of-the-art inner approximations in terms of computational complexity and accuracy for different objectives. Moreover, we propose an efficient disaggregation method applicable to any vertex-based approximation. The proposed methods provide an efficient means to aggregate and to disaggregate typical battery storages in quarter-hourly periods over an entire day with reasonable accuracy for aggregated cost and for peak power optimization.

Stress testing is part of today’s bank risk management and often required by the governing regulatory authority. Performing such a stress test with stress scenarios derived from a distribution, instead of pre-defined expert scenarios, results in a systematic approach in which new severe scenarios can be discovered. The required scenario distribution is obtained from historical time series via a Vector-Autoregressive time series model. The worst-case search, i.e. finding the scenario yielding the most severe situation for the bank, can be stated as an optimization problem. The problem itself is a constrained optimization problem in a high-dimensional search space. The constraints are the box constraints on the scenario variables and the plausibility of a scenario.
The latter is expressed by an elliptic constraint. As the evaluation of the stress scenarios is performed with a simulation tool, the optimization problem can be seen as black-box optimization problem. Evolution Strategy, a well-known optimizer for black-box problems, is applied here. The necessary adaptations to the algorithm are explained and a set of different algorithm design choices are investigated. It is shown that a simple box constraint handling method, i.e. setting variables which violate a box constraint to the respective boundary of the feasible domain, in combination with a repair of implausible scenarios provides good results.

In engineering design, optimization methods are frequently used to improve the initial design of a product. However, the selection of an appropriate method is challenging since many
methods exist, especially for the case of simulation-based optimization. This paper proposes a systematic procedure to support this selection process. Building upon quality function deployment, end-user and design use case requirements can be systematically taken into account via a decision
matrix. The design and construction of the decision matrix are explained in detail. The proposed
procedure is validated by two engineering optimization problems arising within the design of box-type boom cranes. For each problem, the problem statement and the respectively applied optimization methods are explained in detail. The results obtained by optimization validate the use
of optimization approaches within the design process. The application of the decision matrix shows the successful incorporation of customer requirements to the algorithm selection.