In interference-limited wireless networks, interference management techniques are important in order to improve the performance of the systems. Given that spectrum and energy are scarce resources in these networks, techniques that exploit the resources efficiently are desired. We consider a set of base stations operating concurrently in the same spectral band. Each base station is equipped with multiple antennas and transmits data to a single-antenna mobile user. This setting corresponds to the multiple-input single-output (MISO) interference channel (IFC). The receivers are assumed to treat interference signals as noise. Moreover, each transmitter is assumed to know the channels between itself and all receivers perfectly. We study the conflict between the transmitter-receiver pairs (links) using models from game theory and microeconomic theory. These models provide solutions to resource allocation problems which in our case correspond to the joint beamforming design at the transmitters. Our interest lies in solutions that are Pareto optimal. Pareto optimality ensures that it is not further possible to improve the performance of any link without reducing the performance of another link. Strategic games in game theory determine the noncooperative choice of strategies of the players. The outcome of a strategic game is a Nash equilibrium. While the Nash equilibrium in the MISO IFC is generally not efficient, we characterize the necessary null-shaping constraints on the strategy space of each transmitter such that the Nash equilibrium outcome is Pareto optimal. An arbitrator is involved in this setting which dictates the constraints at each transmitter. In contrast to strategic games, coalitional games provide cooperative solutions between the players. We study cooperation between the links via coalitional games without transferable utility. Cooperative beamforming schemes considered are either zero forcing transmission or Wiener filter precoding. We characterize the necessary and sufficient conditions under which the core of the coalitional game with zero forcing transmission is not empty. The core solution concept specifies the strategies with which all players have the incentive to cooperate jointly in a grand coalition. While the core only considers the formation of the grand coalition, coalition formation games study coalition dynamics. We utilize a coalition formation algorithm, called merge-and-split, to determine stable link grouping. Numerical results show that while in the low signal-to-noise ratio (SNR) regime noncooperation between the links is efficient, at high SNR all links benefit in forming a grand coalition. Coalition formation shows its significance in the mid SNR regime where subset link cooperation provides joint performance gains. We use the models of exchange and competitive market from microeconomic theory to determine Pareto optimal equilibria in the two-user MISO IFC. In the exchange model, the links are represented as consumers that can trade goods within themselves. The goods in our setting correspond to the parameters of the beamforming vectors necessary to achieve all Pareto optimal points in the utility region. We utilize the conflict representation of the consumers in the Edgeworth box, a graphical tool that depicts the allocation of the goods for the two consumers, to provide closed-form solution to all Pareto optimal outcomes. The exchange equilibria are a subset of the points on the Pareto boundary at which both consumers achieve larger utility then at the Nash equilibrium. We propose a decentralized bargaining process between the consumers which starts at the Nash equilibrium and ends at an outcome arbitrarily close to an exchange equilibrium. The design of the bargaining process relies on a systematic study of the allocations in the Edgeworth box. In comparison to the exchange model, a competitive market additionally defines prices for the goods. The equilibrium in this economy is called Walrasian and corresponds to the prices that equate the demand to the supply of goods. We calculate the unique Walrasian equilibrium and propose a coordination process that is realized by the arbitrator which distributes the Walrasian prices to the consumers. The consumers then calculate in a decentralized manner their optimal demand corresponding to beamforming vectors that achieve the Walrasian equilibrium. This outcome is Pareto optimal and lies in the set of exchange equilibria. In this thesis, based on the game theoretic and microeconomic models, efficient beamforming strategies are proposed that jointly improve the performance of the systems. The gained results are applicable in interference-limited wireless networks requiring either coordination from the arbitrator or direct cooperation between the transmitters.

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In interference-limited wireless networks, interference management techniques are important in order to improve the performance of the systems. Given that spectrum and energy are scarce resources in these networks, techniques that exploit the resources efficiently are desired. We consider a set of base stations operating concurrently in the same spectral band. Each base station is equipped with multiple antennas and transmits data to a single-antenna mobile user. This setting corresponds to the multiple-input single-output (MISO) interference channel (IFC). The receivers are assumed to treat interference signals as noise. Moreover, each transmitter is assumed to know the channels between itself and all receivers perfectly. We study the conflict between the transmitter-receiver pairs (links) using models from game theory and microeconomic theory. These models provide solutions to resource allocation problems which in our case correspond to the joint beamforming design at the transmitters. Our interest lies in solutions that are Pareto optimal. Pareto optimality ensures that it is not further possible to improve the performance of any link without reducing the performance of another link. Strategic games in game theory determine the noncooperative choice of strategies of the players. The outcome of a strategic game is a Nash equilibrium. While the Nash equilibrium in the MISO IFC is generally not efficient, we characterize the necessary null-shaping constraints on the strategy space of each transmitter such that the Nash equilibrium outcome is Pareto optimal. An arbitrator is involved in this setting which dictates the constraints at each transmitter. In contrast to strategic games, coalitional games provide cooperative solutions between the players. We study cooperation between the links via coalitional games without transferable utility. Cooperative beamforming schemes considered are either zero forcing transmission or Wiener filter precoding. We characterize the necessary and sufficient conditions under which the core of the coalitional game with zero forcing transmission is not empty. The core solution concept specifies the strategies with which all players have the incentive to cooperate jointly in a grand coalition. While the core only considers the formation of the grand coalition, coalition formation games study coalition dynamics. We utilize a coalition formation algorithm, called merge-and-split, to determine stable link grouping. Numerical results show that while in the low signal-to-noise ratio (SNR) regime noncooperation between the links is efficient, at high SNR all links benefit in forming a grand coalition. Coalition formation shows its significance in the mid SNR regime where subset link cooperation provides joint performance gains. We use the models of exchange and competitive market from microeconomic theory to determine Pareto optimal equilibria in the two-user MISO IFC. In the exchange model, the links are represented as consumers that can trade goods within themselves. The goods in our setting correspond to the parameters of the beamforming vectors necessary to achieve all Pareto optimal points in the utility region. We utilize the conflict representation of the consumers in the Edgeworth box, a graphical tool that depicts the allocation of the goods for the two consumers, to provide closed-form solution to all Pareto optimal outcomes. The exchange equilibria are a subset of the points on the Pareto boundary at which both consumers achieve larger utility then at the Nash equilibrium. We propose a decentralized bargaining process between the consumers which starts at the Nash equilibrium and ends at an outcome arbitrarily close to an exchange equilibrium. The design of the bargaining process relies on a systematic study of the allocations in the Edgeworth box. In comparison to the exchange model, a competitive market additionally defines prices for the goods. The equilibrium in this economy is called Walrasian and corresponds to the prices that equate the demand to the supply of goods. We calculate the unique Walrasian equilibrium and propose a coordination process that is realized by the arbitrator which distributes the Walrasian prices to the consumers. The consumers then calculate in a decentralized manner their optimal demand corresponding to beamforming vectors that achieve the Walrasian equilibrium. This outcome is Pareto optimal and lies in the set of exchange equilibria. In this thesis, based on the game theoretic and microeconomic models, efficient beamforming strategies are proposed that jointly improve the performance of the systems. The gained results are applicable in interference-limited wireless networks requiring either coordination from the arbitrator or direct cooperation between the transmitters.

Der zunehmende Einsatz regenerativer Energiequellen erfordert die Nutzung von Energiezwischenspeichern, die umweltfreundlich, günstig und skalierbar sein sollten. Die Zink-Luft-Flussbatterie (ZLFB) kann perspektivisch diese Anforderungen erfüllen, wobei zur Bereitstellung der gespeicherten Energie eine Suspension aus Zinkpartikeln in einem gelierten Elektrolyt durch eine elektrochemische Zelle gepumpt wird. Um die Strömungsstruktur der ZLFB auszulegen und Fehlfunktionen zu vermeiden, ist ein grundlegendes Verständnis der Rheologie der Zinksuspension notwendig. Außerdem kann über die Einstellung einer geeigneten Strömung die bei der Entladung erreichte elektrische Leistungsdichte gesteigert werden. Bereits die Flüssigphase der Zinksuspension weist eine komplexe nicht-Newtonsche Rheologie auf, welche durch die Zugabe der Partikel komplexer wird. Für das grundlegende Verständnis der Rheologie werden daher Modellexperimente durchgeführt, wobei in dieser Arbeit ein L-förmiger Kanal mit Strömungsaufweitung untersucht wurde, um die komplexen strömungsmechanischen Eigenschaften der Zinksuspension abzubilden. Zur Erfassung des Strömungsfeldes ist eine Ortsauflösung von 1 … 2 mm in einem Messbereich von 20 × 15 mm2 erforderlich. Ultraschall ist prinzipiell geeignet, um das Strömungsfeld in der opaken Suspension zu erfassen, wobei die wesentliche Herausforderung in den starken Wellenfrontverzerrungen besteht, welche durch die Zinkpartikel eingebracht werden. Es konnte gezeigt werden, dass die Ultrasound Imaging Velocimetry (UIV) robuster gegenüber diesen Störungen ist, als die Ultraschall-Doppler-Velozimetrie (UDV). Die UIV wurde daher mittels Geschwindigkeitsnormal an die messtechnischen Randbedingungen der Zinksuspension angepasst und charakterisiert. Bei einer Ortsauflösung von 1,6 mm wurde eine Gesamtmessunsicherheit von 2,5 % axial und 4,1 % lateral zur Schallausbreitungsrichtung erreicht. Das im Modellexperiment gemessene Strömungsfeld weist eine Totzone an der Strömungsumlenkung auf, deren Auftreten durch eine von der Scherhistorie abhängige Viskosität erklärt werden kann. Dieser Effekt wird als Thixotropie bezeichnet. Durch die In-situ-Messung der Strömung in einer aktiven ZLFB kann eine Korrelation von Strömung und elektrischer Leistung erfolgen und die erzielte Leistungsdichte perspektivisch durch eine Anpassung der Strömung gesteigert werden. Bei der Messung im 2,6 mm hohen Anodenspalt muss aufgrund der komplexen Rheologie der Suspension und der daraus resultierenden hohen Geschwindigkeitsgradienten eine Ortsauflösung von unter 100 µm bei gleichzeitig kleiner Ultraschallfrequenz realisiert werden, da der Ultraschall für hohe Frequenzen nicht in die Zinksuspension eindringt. Um dieses Ziel zu erreichen, wurde die Super Resolution Ultrasound Particle Tracking Velocimetry (SRPTV) genutzt, welche Ortsauflösungen unterhalb des Beugungslimits ermöglicht. Einzelne nichtlineare Streupartikel werden mittels Harmonic Imaging isoliert abgebildet und verfolgt, wobei die durch die Zinkpartikel eingebrachten Wellenfrontstörungen durch einen kohärenzgewichteten Strahlformer kompensiert werden. Es wurde eine Ortsauflösung von 67 µm axial und 30 µm lateral zur Schallausbreitungsrichtung bei einer Anregungswellenlänge von 330 µm erreicht. Trotz der stark streuenden Zinksuspension, konnte so eine Messung in der aktiven ZLFB mit einer maximalen Messunsicherheit von 12,5 % durchgeführt werden. Dabei wurde eine Wandgleitgeschwindigkeit von 3 mm s−1 bei einer maximalen Geschwindigkeit von etwa 8 mm s−1 festgestellt. Die SRPTV kann darüber hinaus in anderen technischen Prozessen eingesetzt werden, in denen Suspensionsströmungen in kleinen Geometrien auftreten.:Symbolverzeichnis xiii Abkürzungsverzeichnis xv 1 Einleitung 1 1.1 Motivation und Zielstellung 1 1.2 Stand der Technik 4 1.3 Lösungsansatz und Struktur der Arbeit 8 2 Theoretische Grundlagen 11 2.1 Grundlagen der Schallausbreitung 11 2.1.1 Schallausbreitung in homogenen Medien 11 2.1.2 Schallausbreitung in inhomogenen Medien 13 2.2 Ultraschall Bildgebung 14 2.2.1 Phased-Array-Prinzip 15 2.2.2 Plane wave imaging 16 2.2.3 Grenzen der Schallfeldfokussierung 16 2.3 Messung von Strömungsfeldern 17 2.3.1 Ultraschall-Doppler-Velozimetrie 18 2.3.2 Ultrasound Imaging Velocimetry 19 2.3.3 Ultrasound Particle Tracking Velocimetry 19 2.4 Nichtlineare akustische Effekte 20 2.4.1 Beschreibung von Linearität 20 2.4.2 Interaktion von nichtlinearen Streupartikeln und Schallwelle 20 2.4.3 Harmonic Imaging 21 3 Experimentelle Grundlagen 25 3.1 Charakterisierung der Zinksuspension 25 3.1.1 Zusammensetzung der Zinksuspension 25 3.1.2 Bestimmung von Dämpfung und Schallgeschwindigkeit 26 3.1.3 Bestimmung der spezifischen akustischen Impedanz 29 3.2 Messtechnik 30 3.2.1 Ultraschallforschungsplattform: Phased Array Ultrasound Dopp- ler Velocimeter 30 3.2.2 Ultraschallwandler 32 4 Verfahren zur Strömungsmessung im Modellexperiment 37 4.1 Experimenteller Aufbau 37 4.2 Untersuchung geeigneter Verfahren zur Messung von Strömungsfel- dern in der Zinksuspension 38 4.3 Optimierung der Signalverarbeitung und Charakterisierung der Messei- genschaften 44 4.3.1 Geschwindigkeitsnormal 44 4.3.2 Optimierung der Messsystemparameter 45 4.3.3 Charakterisierung der Messeigenschaften 48 4.3.4 Validierung 49 4.4 Messung der Suspensionsströmung im Modellexperiment 55 4.4.1 Messergebnisse 55 4.4.2 Vergleich von Simulation und Messung 58 4.5 Fazit 61 5 Verfahren zur In-situ-Strömungsmessung in einer Zink-Luft-Flussbatterie 63 5.1 Experimenteller Aufbau 63 5.2 Strömungsmessung unterhalb des Beugungslimits - Super Resolution Ultrasound Particle Tracking Velocimetry (SRPTV) 65 5.2.1 Nutzung nichtlinearer Streupartikel 68 5.2.2 Trennung von linearem und nichtlinearem Signalanteil 72 5.2.3 Strahlformung mit Kompensation der Streuung 76 5.2.4 Particle Tracking 80 5.3 Charakterisierung der Messeigenschaften 81 5.3.1 Vorgehen zur Charakterisierung der Messeigenschaften 82 5.3.2 Untersuchung der Positionsunsicherheit 83 5.3.3 Untersuchung der Geschwindigkeitsunsicherheit 92 5.4 Messung an einer aktiven Zink-Luft-Flussbatterie 95 5.4.1 Aufbau und Durchführung 95 5.4.2 Messergebnisse 97 5.4.3 Vergleich von Simulation und Messung 97 5.5 Fazit 102 6 Zusammenfassung und Ausblick 103 6.1 Erkenntnisse und Fortschritt 103 6.1.1 Ultrasound Imaging Velocimetry 103 6.1.2 Super Resolution Ultrasound Particle Tracking Velocimetry 104 6.1.3 Fazit 106 6.2 Ausblick und weiterführende Arbeiten 106 6.2.1 Messtechnik 106 6.2.2 Anwendung 107 Literaturverzeichnis 109 Publikationsverzeichnis 117 Artikel in Zeitschriften mit peer-review 117 Tagungsbeiträge 117 Patente 119 / For the efficient use of renewable energies, energy storage systems are required that are environmentally friendly, low priced and scalable. The zinc-air flow battery (ZAB), which is operated by pumping an opaque suspension of zinc particles in an gelled electrolyte through an electrochemical cell, is a promising candidate as energy storage system for these requirements. To design the fluidic structures and avoid malfunction, a fundamental understanding of the rheology of the zinc suspension is required. Additionally, the electrical performance of the cell can be imporved by optimizing the flow in the electrochemical cell. The liquid phase of the suspension itself has complex non-Newtonian properties, which are even more complex when the particles are considered. For the fundamental understanding of the suspension rheology, model experiments are conducted. In this work an L-shaped channel with a widening is used to represent relevant effects from the complex rheology of the suspension. To measure the flow field, a spatial resolution of 1 … 2 mm and a measurement area of 20 × 15 mm2 are required. Ultrasound can be used to measure the flow in opaque liquids, but wavefront distortions are introduced by the zinc particles. Established measurement methods for homogeneous opaque fluids, the Ultrasound Imaging Velocimetry (UIV) and the Ultrasound Doppler Velocimetry (UDV), were compared for the application at the suspension. The UIV has a 50 % lower random deviation, which makes it more suitable for the flow measurement in the suspension and it was adapted to the measurement conditions in the suspension. At a spatial resolution of 1.66 mm, a velocity uncertainty of 2.5 % axial and 4.1 % lateral to the ultrasound propagation were achieved. The application of the UIV to the suspension flow in the model experiment revealed a thixotropic behavior of the fluid, which resulted in a dead flow zone opposite to the inlet of the channel. The in situ measurement of the flow in an active ZAB, allows to correlate electrical performance and flow and thereby an improvement of the cell performance by adapting the flow. For the measurement in the anodic channel with a width of 2.6 mm, a spatial resolution of 100 µm is required because of the high velocity gradients due to the non-Newtonian rheology of the suspension. The high spatial resolution has to be achieved at low ultrasound frequencies, since the ultrasound does not penetrate into the suspension for high frequencies. To achieve this, the Super Resolution Ultrasound Particle Tracking Velocimetry (SRPTV) was used, which allows a spatial resolution beyond the diffraction limit. Harmonic Imaging is used to image isolated non-linear tracer particles, which are tracked for velocity estimation. The speckle and image distortion due to the induced wavefront distortions are compensated with a coherence weighting beamformer. A spatial resolution of 67 µm axial and 30 µm lateral to the ultrasound propagation were achieved. Despite the strong scattering of the ultrasound at the zinc particles, a maximum velocity uncertainty of 12.5 % referred to the maximum velocity was achieved for the measurement in the active ZAB. A slip velocity of 3 mm at a maximum velocity of 8 mm was observed. The SRPTV can be applied to other technical processes, where suspension flows in small geometries play an important role.:Symbolverzeichnis xiii Abkürzungsverzeichnis xv 1 Einleitung 1 1.1 Motivation und Zielstellung 1 1.2 Stand der Technik 4 1.3 Lösungsansatz und Struktur der Arbeit 8 2 Theoretische Grundlagen 11 2.1 Grundlagen der Schallausbreitung 11 2.1.1 Schallausbreitung in homogenen Medien 11 2.1.2 Schallausbreitung in inhomogenen Medien 13 2.2 Ultraschall Bildgebung 14 2.2.1 Phased-Array-Prinzip 15 2.2.2 Plane wave imaging 16 2.2.3 Grenzen der Schallfeldfokussierung 16 2.3 Messung von Strömungsfeldern 17 2.3.1 Ultraschall-Doppler-Velozimetrie 18 2.3.2 Ultrasound Imaging Velocimetry 19 2.3.3 Ultrasound Particle Tracking Velocimetry 19 2.4 Nichtlineare akustische Effekte 20 2.4.1 Beschreibung von Linearität 20 2.4.2 Interaktion von nichtlinearen Streupartikeln und Schallwelle 20 2.4.3 Harmonic Imaging 21 3 Experimentelle Grundlagen 25 3.1 Charakterisierung der Zinksuspension 25 3.1.1 Zusammensetzung der Zinksuspension 25 3.1.2 Bestimmung von Dämpfung und Schallgeschwindigkeit 26 3.1.3 Bestimmung der spezifischen akustischen Impedanz 29 3.2 Messtechnik 30 3.2.1 Ultraschallforschungsplattform: Phased Array Ultrasound Dopp- ler Velocimeter 30 3.2.2 Ultraschallwandler 32 4 Verfahren zur Strömungsmessung im Modellexperiment 37 4.1 Experimenteller Aufbau 37 4.2 Untersuchung geeigneter Verfahren zur Messung von Strömungsfel- dern in der Zinksuspension 38 4.3 Optimierung der Signalverarbeitung und Charakterisierung der Messei- genschaften 44 4.3.1 Geschwindigkeitsnormal 44 4.3.2 Optimierung der Messsystemparameter 45 4.3.3 Charakterisierung der Messeigenschaften 48 4.3.4 Validierung 49 4.4 Messung der Suspensionsströmung im Modellexperiment 55 4.4.1 Messergebnisse 55 4.4.2 Vergleich von Simulation und Messung 58 4.5 Fazit 61 5 Verfahren zur In-situ-Strömungsmessung in einer Zink-Luft-Flussbatterie 63 5.1 Experimenteller Aufbau 63 5.2 Strömungsmessung unterhalb des Beugungslimits - Super Resolution Ultrasound Particle Tracking Velocimetry (SRPTV) 65 5.2.1 Nutzung nichtlinearer Streupartikel 68 5.2.2 Trennung von linearem und nichtlinearem Signalanteil 72 5.2.3 Strahlformung mit Kompensation der Streuung 76 5.2.4 Particle Tracking 80 5.3 Charakterisierung der Messeigenschaften 81 5.3.1 Vorgehen zur Charakterisierung der Messeigenschaften 82 5.3.2 Untersuchung der Positionsunsicherheit 83 5.3.3 Untersuchung der Geschwindigkeitsunsicherheit 92 5.4 Messung an einer aktiven Zink-Luft-Flussbatterie 95 5.4.1 Aufbau und Durchführung 95 5.4.2 Messergebnisse 97 5.4.3 Vergleich von Simulation und Messung 97 5.5 Fazit 102 6 Zusammenfassung und Ausblick 103 6.1 Erkenntnisse und Fortschritt 103 6.1.1 Ultrasound Imaging Velocimetry 103 6.1.2 Super Resolution Ultrasound Particle Tracking Velocimetry 104 6.1.3 Fazit 106 6.2 Ausblick und weiterführende Arbeiten 106 6.2.1 Messtechnik 106 6.2.2 Anwendung 107 Literaturverzeichnis 109 Publikationsverzeichnis 117 Artikel in Zeitschriften mit peer-review 117 Tagungsbeiträge 117 Patente 119