Zulässigkeit für maximale Unsicherheitsmengen in robuster Optimierung mit Anwendung in Gasnetzen

Robust optimization is a popular approach to protect an optimization problem against uncertain data within a user-specified set of scenarios, the so-called uncertainty set. In many cases, the choice of the uncertainty set is driven by the application. In general, it can be elusive to assume that the exact “size” of the uncertainty set can be specified prior to the optimization process. Overly large sized uncertainty sets can lead to infeasible robust optimization problems. To avoid robust infeasibility due to the choice of the uncertainty set, it is useful to know the maximal “size” of a given uncertainty set such that feasibility of the robust optimization problem is still guaranteed. We study maximal uncertainty sets that guarantee robust feasibility for general mixed-integer linear problems (MIPs) and in the context of gas networks in this cumulative dissertation. In the first part, we summarize and discuss our results developed over the last years. The second part of this cumulative dissertation contains reprints of our original articles and preprints, which contain all details of the presented results. We also refer to these articles throughout the first part of this dissertation. For general MIPs, we consider a specific notion for the maximal size of a given uncertainty set: the radius of robust feasibility (RRF). We introduce and study the RRF for MIPs under common assumptions from the literature and then extend the RRF to include “safe” variables and constraint, i.e., variables and constraints that are not affected by uncertainties. We further develop methods for computing the RRF of linear and mixed-integer linear problems with safe variables and constraints and successfully apply them to instances of the MIPLIB 2017 library. Based on our results, we can control the price of robustness by adjusting the size of the uncertainty set. Moreover, we study the two-stage robust problems of deciding the feasibility of a booking as well as of computing maximal technical capacities within the European entry-exit gas market system. A booking is a capacity-right contract for which the transmission system operator has to guarantee that every balanced load flow below the booking can be transported through the network. Maximal technical capacities bound these bookings and, thus, describe maximal bookable capacities. Except for some technical subtleties, these robust problems lead to deciding the feasibility as well as solving a specific two-stage robust nonlinear optimization problem. The main goal of this problem consists of computing a maximal uncertainty set of balanced load flows so that for each of these load flows there is a feasible transport through the network. We study this problem algorithmically with focus on nonlinear models of gas transport. We analyze structural properties such as (non-)convexity of the set of feasible bookings and of the set of feasible balanced load flows for different models of gas transport. For deciding the feasibility of a booking, we develop a polynomial-time algorithm for single-cycle networks consisting of pipes. We also characterize feasible bookings in networks with compressors and control valves. Based on the results for bookings, we provide results for computing maximal technical capacities in tree-shaped networks. We note that our results can also contribute to other potential-based network problems such as computing a robust diameter selection for tree-shaped hydrogen networks with demand uncertainties.Viele Entscheidungen im realen Leben basieren aus unterschiedlichen Gründen auf unsicheren Daten. Eine mögliche Quelle für Unsicherheiten können Abweichungen in Vorhersagen und Prognosen, beispielsweise für den zukünftigen Bedarf von Wasserstoff, sein. Unsicherheiten in den Daten führen zu unsicheren Parametern in Optimierungsproblemen, die häufig reale Entscheidungsprozesse unterstützen. Die Berücksichtigung von Unsicherheiten in Optimierungsproblemen ist von großer Bedeutung, da bereits kleine Störungen in den Daten zu suboptimalen oder sogar unzulässigen Lösungen führen können. In dieser kumulativen Dissertation fokussieren wir uns auf den etablierten Ansatz der robusten Optimierung, um Unsicherheiten in Optimierungsproblemen zu berücksichtigen. Das Hauptziel der robusten Optimierung ist die Berechnung einer Lösung, die zulässig ist für alle – gewöhnlich unendlich viele – Unsicherheiten innerhalb einer vorgegebenen Unsicherheitsmenge und optimal unter dieser Bedingung ist. Ein Großteil der Literatur behandelt die Berechnung einer solchen robusten Lösung. Die Wahl der gegebenen Unsicherheitsmenge ist vergleichsweise wenig untersucht. Häufig wird die Wahl der Unsicherheitsmenge durch anwendungsorientierte Aspekte bestimmt. Dennoch ist es im Allgemeinen nicht zu erwarten, dass die exakte „Größe“ der Unsicherheitsmenge vor dem Optimierungsprozess bekannt ist. Zu groß gewählte Unsicherheitsmengen können zu unzulässigen robusten Optimierungsproblemen führen. Um robuste Unzulässigkeit auf Grund der Wahl der Unsicherheitsmenge zu vermeiden, ist es hilfreich, die maximale „Größe“ einer gegebenen Unsicherheitsmenge zu kennen, sodass mindestens eine robuste Lösung existiert. In dieser kumulativen Dissertation untersuchen wir maximale Unsicherheitsmengen, die Zulässigkeit für robuste Optimierungsprobleme garantieren, sowohl für gemischt-ganzzahlige lineare Probleme als auch im Kontext von Gasnetzen. Im ersten Teil dieser kumulativen Dissertation fassen wir unsere erzielten Ergebnisse bezüglich maximaler Unsicherheitsmengen mit Zulässigkeitsgarantie zusammen und diskutieren die Resultate. Der zweite Teil beinhaltet die reproduzierten Publikationen und Vorveröffentlichungen, die alle Details zu unseren vorgestellten Ergebnissen enthalten. Die jeweiligen Beiträge des Autors dieser kumulativen Dissertation zu diesen Artikeln sind im Abschnitt „Author’s Contributions“ ab Seite ix dargelegt. Nachfolgend beschreiben wir kurz den Inhalt des ersten Teils der Dissertation. Für gemischt-ganzzahlige lineare Optimierungsprobleme betrachten wir ein bestimmtes Maß für die Größe der Unsicherheitsmenge: „den Radius der robusten Zulässigkeit/radius of robust feasibility“ (RRF). Wir führen den RRF für gemischt-ganzzahlige lineare Optimierungsprobleme ein und analysieren den RRF eines gemischt-ganzzahligen linearen Problems und seiner kontinuierlichen Relaxierung unter den üblichen Annahmen der Literatur. Anschließend erweitern wir den RRF um „sichere“ Variablen und Nebenbedingungen, das heißt Variablen und Nebenbedingungen, die nicht von Unsicherheiten betroffen sind. Wir entwickeln Lösungsmethoden, die den RRF mit sicheren Variablen und Nebenbedingungen berechnen, und wenden diese in einer ausführlichen numerischen Studie an. Mit Hilfe unserer Ergebnisse können wir den „Preis der Robustheit“ durch Adjustieren der Größe der Unsicherheitsmenge kontrollieren. Zusätzlich untersuchen wir die zweistufigen robusten Probleme der Buchungsvalidierung und der Berechnung von maximalen technischen Kapazitäten im europäischen Entry-Exit Gasmarktsystem. Eine Buchung ist ein Vertrag zwischen Gashändlern und dem Fernleitungsnetzbetreiber (FNB) bezüglich Ein- und Ausspeisekapazitäten im Gasnetz. Hierbei garantiert der FNB, dass jeder balancierte Lastfluss innerhalb der Buchung im Netz transportiert werden kann. Technische Kapazitäten sind Knoten-kapazitäten, die die Buchungen begrenzen und somit maximal buchbare Kapazitäten ausweisen. Bis auf technische Feinheiten entsprechen die Buchungsvalidierung und die Berechnung maximaler technischer Kapazitäten der Entscheidung über Zulässigkeit beziehungsweise dem Lösen eines spezifischen zweistufigen robusten nichtlinearen Optimierungsproblems. Das Hauptziel dieses robusten Problems besteht in der Berechnung einer maximalen Unsicherheitsmenge von balancierten Lastflüssen, sodass jeder dieser Lastflüsse im Netz transportiert werden kann. Wir untersuchen das betrachtete zweistufige robuste Optimierungsproblem algorithmisch mit Fokus auf nichtlinearen Modellierungen des Gastransports. Dabei betrachten wir passive Netze, die nur aus Rohren bestehen, und aktive Netze, die zusätzlich aktiv steuerbare Kompressoren und Regler enthalten. Wir analysieren strukturelle Eigenschaften, beispielsweise (Nicht-)Konvexität, der Menge der zulässigen Buchungen als auch der Menge der zulässigen balancierten Lastflüsse für verschiedene Modellierungen des Gasflusses. Aus der Literatur ist bekannt, dass die Zulässigkeit einer Buchung in polynomieller Zeit für passive Netze mit Baumstruktur entschieden werden kann. Dieses Resultat kann auch mit Hilfe unserer Ergebnisse, basierend auf einer leicht unterschiedlichen Herangehensweise, gezeigt werden. Wir entwickeln einen Algorithmus, der die Zulässigkeit einer Buchung für passive Netze bestehend aus einem Kreis in polynomieller Zeit entscheidet. Die Buchungsvalidierung auf allgemeinen passiven Netzen ist coNP-schwer. Zusätzlich charakterisieren wir zulässige Buchungen für Netze mit Kompressoren und Reglern. Basierend auf den Ergebnissen für Buchungen erzielen wir Resultate zur Berechnung maximaler technischer Kapazitäten in passiven Netzen mit Baumstruktur. Diese Ergebnisse ermöglichen es, ein mehrstufiges Modell des europäischen Entry-Exit Gasmarktes aus der Literatur für passive Netze mit Baumstruktur in realer Größe und einem nichtlinearen Modell für den Gastransport zu lösen. Abschließend weisen wir darauf hin, dass unsere Ergebnisse auch zu anderen potenzialbasierten Netzwerkproblemen beitragen können, wie zum Beispiel der Berechnung einer robusten Durchmesserauswahl für baumförmige Wasserstoffnetze mit Bedarfsunsicherheiten

Computing technical capacities in the European entry-exit gas market is NP-hard

Abstract As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes of the network. Second, the TSO and the gas traders sign mid- to long-term capacity-right contracts, where the capacity is bounded above by the allocated technical capacities. These contracts are called bookings. Third, on a day-ahead basis, gas traders can nominate the amount of gas that they inject or withdraw from the network at entry and exit nodes, where the nominated amount is bounded above by the respective booking. Fourth and finally, the TSO has to operate the network such that the nominated amounts of gas can be transported. By signing the booking contract, the TSO guarantees that all possibly resulting nominations can indeed be transported. Consequently, maximal technical capacities have to satisfy that all nominations that comply with these technical capacities can be transported through the network. This leads to a highly challenging mathematical optimization problem. We consider the specific instantiations of this problem in which we assume capacitated linear as well as potential-based flow models. In this contribution, we formally introduce the problem of Computing Technical Capacities (CTC) and prove that it is NP-complete on trees and NP-hard in general. To this end, we first reduce the Subset Sum problem to CTC for the case of capacitated linear flows in trees. Afterward, we extend this result to CTC with potential-based flows and show that this problem is also NP-complete on trees by reducing it to the case of capacitated linear flow. Since the hardness results are obtained for the easiest case, i.e., on tree-shaped networks with capacitated linear as well as potential-based flows, this implies the hardness of CTC for more general graph classes

Bilevel Optimization Approaches to Decide the Feasibility of Bookings in the European Gas Market

The European gas market is organized as a so-called entry-exit system with the main goal to decouple transport and trading. To this end, gas traders and the transmission system operator (TSO) sign so-called booking contracts that grant capacity rights to traders to inject or withdraw gas at certain nodes up to this capacity. On a day-ahead basis, traders then nominate the actual amount of gas within the previously booked capacities. By signing a booking contract, the TSO guarantees that all nominations within the booking bounds can be transported through the network. This results in a highly challenging mathematical problem. Using potential-based flows to model stationary gas physics, feasible bookings on passive networks, i.e., networks without controllable elements, have been characterized in the recent literature. In this paper, we consider networks with linearly modeled active elements such as compressors and control valves that do not lie on cycles of the network. Since these active elements allow the TSO to control the gas flow, the single-level approaches from the literature are no longer applicable. We thus present a bilevel approach to decide the feasibility of bookings in networks with active elements. Besides the classical Karush-Kuhn-Tucker reformulation, we obtain three problem-specific optimal-value-function reformulations, which also lead to novel characterizations of feasible bookings in active networks. We compare the performance of our methods by a case study based on data from the GasLib

Successful Pharmacologic Treatment of Self-Bloodletting with Factitious Chronic Anemia (Lasthénie de Ferjol Syndrome) with High-Dose Serotonergic Medication: A Case Report

Self-induced bloodletting (SBL) is a very rare form of self-injury (SI) seen primarily in adolescents and young adults with personality and eating disorders. It can result in complications like malaise, fatigue, or iron-deficiency anemia (Lasthénie de Ferjol syndrome, LFS), and poses a risk of accidental death or suicide. The condition often goes undetected due to patient concealment. There is no specific treatment established, and pharmacological strategies remain uncertain. We discuss the case of a 22-year-old female patient treated at our Psychiatry and Psychotherapy Department following a suicide attempt via SBL. She self-administered a venous cannula, losing 1.5 L of blood. Diagnosed with iron-deficiency anemia (LFS), she was initially treated with mirtazapine, risperidone, lithium, and later off-label high-dose clomipramine (300 mg/d). Clomipramine significantly reduced her SBL and suicidal thoughts, and her hemoglobin levels re-normalized under iron-substitution therapy. Despite improvement and later discharge, she attempted suicide by SBL again three months later, having stopped clomipramine due to adverse side effects. High-dose escitalopram was administered, leading to a decrease and eventual cessation of her SBL urges. This case demonstrates that patients with SBL/LFS can benefit from high-dose clomipramine or escitalopram. Despite its rarity, the consideration of high-dose serotonergic antidepressants is crucial in psychiatric diagnostics and treatment for patients affected by SBL/LFS.</p

Modeling Hydrogen Networks for Future Energy Systems: A Comparison of Linear and Nonlinear Approaches

Common energy system models that integrate hydrogen transport in pipelines typically simplify fluid flow models and reduce the network size in order to achieve solutions quickly. This contribution analyzes two different types of pipeline network topologies (namely, star and tree networks) and two different fluid flow models (linear and nonlinear) for a given hydrogen capacity scenario of electrical reconversion in Germany to analyze the impact of these simplifications. For each network topology, robust demand and supply scenarios are generated. The results show that a simplified topology, as well as the consideration of detailed fluid flow, could heavily influence the total pipeline investment costs. For the given capacity scenario, an overall cost reduction of the pipeline costs of 37% is observed for the star network with linear cost compared to the tree network with nonlinear fluid flow. The impact of these improvements regarding the total electricity reconversion costs has led to a cost reduction of 1.4%, which is fairly small. Therefore, the integration of nonlinearities into energy system optimization models is not recommended due to their high computational burden. However, the applied method for generating robust demand and supply scenarios improved the credibility and robustness of the network topology, while the simplified fluid flow consideration can lead to infeasibilities. Thus, we suggest the utilization of the nonlinear model for post-processing to prove the feasibility of the results and strengthen their credibility, while retaining the computational performance of linear modeling

The Effects of Different Isocaloric Oral Nutrient Solutions on Psychophysical, Metabolic, Cognitive, and Olfactory Function in Young Male Subjects

Food intake influences human cognition, olfaction, hunger, and food craving. However, little research has been done in this field to elucidate the effects of different nutrients. Thus, the goal of our study was to investigate the effects of oral ingestion of different nutrient solutions on olfactory, cognitive, metabolic and psychophysical function. Twenty healthy men participated in our study employing a double-blind, cross-over, repeated measurement design. Participants were tested on four different study days. Each day participants received, in randomized order, one of three isocaloric (protein, carbohydrate or fat 600 kcal, 1,500 mL) solutions or a placebo. Olfactory and cognitive tests (monitoring only) were conducted three times, i.e., 60 min before the beginning of nutrient intake, following oral ingestion of the solution and 60, and 240 min after. Psychophysical and metabolic function tests (active grehlin, desacyl ghrelin, insulin, glucagon, glucose, triglyceride, urea) were performed 7 times on each examination day (observation period: −60 min, 0 = solution intake, +60, +120, +180, +240, and +300 min). Ratings of hunger and food craving significantly differed over the observation period with lowest ratings following application of the protein solution. Highest ratings of craving were found following placebo intake. We further observed a significant positive correlation of active grehlin with hunger and fat, protein and sweets craving for each nutrient solution. Active grehlin significantly correlated with carbohydrate craving for carbohydrate and fat solution and with vegetable craving for fat solution only. Hunger hormone levels, hunger and food craving ratings demonstrated that the hierarchical order that appears in satiating efficiencies of isovolumetric-isocaloric ingested macronutrients is protein > fat > carbohydrate. Our study reveals that the type of nutrient exerts a significant influence on metabolic parameters, hunger and food craving

Computing Technical Capacities in the European Entry-Exit Gas Market is NP-Hard

As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes of the network. Second, the TSO and the gas traders sign mid- to long-term capacity-right contracts, where the capacity is bounded above by the allocated technical capacities. These contracts are called bookings. Third, on a day-ahead basis, gas traders can nominate the amount of gas that they inject or withdraw from the network at entry and exit nodes, where the nominated amount is bounded above by the respective booking. Fourth and finally, the TSO has to operate the network such that the nominated amounts of gas can be transported. By signing the booking contract, the TSO guarantees that all possibly resulting nominations can indeed be transported. Consequently, maximal technical capacities have to satisfy that all nominations that comply with these technical capacities can be transported through the network. This leads to a highly challenging mathematical optimization problem. We consider the specific instantiations of this problem in which we assume capacitated linear as well as potential-based flow models. In this contribution, we formally introduce the problem of Computing Technical Capacities (CTC) and prove that it is NP-complete on trees and NP-hard in general. To this end, we first reduce the Subset Sum problem to CTC for the case of capacitated linear flows in trees. Afterward, we extend this result to CTC with potential-based flows and show that this problem is also NP-complete on trees by reducing it to the case of capacitated linear flow. Since the hardness results are obtained for the easiest case, i.e., on tree-shaped networks with capacitated linear as well as potential-based flows, this implies the hardness of CTC for more general graph classes

On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this short note, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to $\varepsilon$-feasibility regarding its nonlinear constraints for an arbitrarily small but positive $\varepsilon$, the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, and for which the convex constraint set satisfies Slater's constraint qualification for all feasible upper-level decisions. Since the consideration of $\varepsilon$-feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we show that the nonlinearities in the lower level are the key reason for the observed bad behavior by proving that this behavior cannot appear for linear bilevel problems