Black holes in the early universe, in compact binaries, and as energy sources inside solar-type stars

This thesis consists of three separate studies of roles that black holes might play in our universe.

In the first part we formulate a statistical method for inferring the cosmological parameters of our universe from LIGO/VIRGO measurements of the gravitational waves produced by coalescing black-hole/neutron-star binaries. This method is based on the cosmological distance-redshift relation, with "luminosity distances" determined directly, and redshifts indirectly, from the gravitational waveforms. Using the current estimates of binary coalescence rates and projected "advanced" LIGO noise spectra, we conclude that by our method the Hubble constant should be measurable to within an error of a few percent. The errors for the mean density of the universe and the cosmological constant will depend strongly on the size of the universe, varying from about 10% for a "small" universe up to and beyond 100% for a "large" universe. We further study the effects of random gravitational lensing and find that it may strongly impair the determination of the cosmological constant.

In the second part of this thesis we disprove a conjecture that black holes cannot form in an early, inflationary era of our universe, because of a quantum-field-theory induced instability of the black-hole horizon. This instability was supposed to arise from the difference in temperatures of any black-hole horizon and the inflationary cosmological horizon; it was thought that this temperature difference would make every quantum state that is regular at the cosmological horizon be singular at the black-hole horizon. We disprove this conjecture by explicitly constructing a quantum vacuum state that is everywhere regular for a massless scalar field. We further show that this quantum state has all the nice thermal properties that one has come to expect of "good" vacuum states, both at the black-hole horizon and at the cosmological horizon.

In the third part of the thesis we study the evolution and implications of a hypothetical primordial black hole that might have found its way into the center of the Sun or any other solar-type star. As a foundation for our analysis, we generalize the mixing-length theory of convection to an optically thick, spherically symmetric accretion flow (and find in passing that the radial stretching of the inflowing fluid elements leads to a modification of the standard Schwarzschild criterion for convection). When the accretion is that of solar matter onto the primordial hole, the rotation of the Sun causes centrifugal hangup of the inflow near the hole, resulting in an "accretion torus" which produces an enhanced outflow of heat. We find, however, that the turbulent viscosity, which accompanies the convective transport of this heat, extracts angular momentum from the inflowing gas, thereby buffering the torus into a lower luminosity than one might have expected. As a result, the solar surface will not be influenced noticeably by the torus's luminosity until at most three days before the Sun is finally devoured by the black hole. As a simple consequence, accretion onto a black hole inside the Sun cannot be an answer to the solar neutrino puzzle.

A tale of two cobalts

There are important problems to overcome if solar energy or other renewable energy sources are to be used effectively on a global scale. Solar photons must not only be harvested and converted into a usable form, but they must also be efficiently stored so that energy is available for use on cloudy days and at night. In this work, both the energy conversion and energy storage problems are addressed. Specifically, two cobalt complexes were designed and their reactivity probed for applications in energy conversion and storage. The first chapter describes a cobalt complex that is the first example of a dimeric cobalt compound with two singly proton-bridged cobaloxime units linked by a central BO4--bridge. Using electrochemical methods, the redox properties of the dimer were evaluated and it was found to be an electrocatalyst for proton reduction in acetonitrile.

Because hydrogen gas is difficult to handle and store, the hydrogenation of CO2 and later dehydrogenation of the liquid product, formic acid, has been proposed as a hydrogen storage system. Thus, a second complex, described in chapter two, supported by a triphosphine ligand framework was used as a catalyst precursor for this key dehydrogenation step. The studies here demonstrate the efficacy of the complex as a precatalyst for the desired reaction, with good conversion of starting formic acid to CO2 and H2. In order to better understand the properties of the triphosphine cobalt complex, a synthetic procedure for substituting electron donating groups (e.g., methoxy groups) onto the ligand was investigated, yielding a novel diphosphine cobalt(II) complex.

Dendrites Inhibition in Rechargeable Lithium Metal Batteries

The specific high energy and power capacities of rechargeable lithium metal (Li⁰) batteries are ideally suited to portable devices and are valuable as storage units for intermittent renewable energy sources. Lithium, the lightest and most electropositive metal, would be the optimal anode material for rechargeable batteries if it were not for the fact that such devices fail unexpectedly by short-circuiting via the dendrites that grow across electrodes upon recharging. This phenomenon poses a major safety issue because it triggers a series of adverse events that start with overheating, potentially followed by the thermal decomposition and ultimately the ignition of the organic solvents used in such devices.

In this thesis, we developed experimental platform for monitoring and quantifying the dendrite populations grown in a Li battery prototype upon charging under various conditions. We explored the effects of pulse charging in the kHz range and temperature on dendrite growth, and also on loss capacity into detached “dead” lithium particles.

Simultaneously, we developed a computational framework for understanding the dynamics of dendrite propagation. The coarse-grained Monte Carlo model assisted us in the interpretation of pulsing experiments, whereas MD calculations provided insights into the mechanism of dendrites thermal relaxation. We also developed a computational framework for measuring the dead lithium crystals from the experimental images.

Sustainable IT and IT for sustainability

Energy and sustainability have become one of the most critical issues of our generation. While the abundant potential of renewable energy such as solar and wind provides a real opportunity for sustainability, their intermittency and uncertainty present a daunting operating challenge. This thesis aims to develop analytical models, deployable algorithms, and real systems to enable efficient integration of renewable energy into complex distributed systems with limited information.

The first thrust of the thesis is to make IT systems more sustainable by facilitating the integration of renewable energy into these systems. IT represents the fastest growing sectors in energy usage and greenhouse gas pollution. Over the last decade there are dramatic improvements in the energy efficiency of IT systems, but the efficiency improvements do not necessarily lead to reduction in energy consumption because more servers are demanded. Further, little effort has been put in making IT more sustainable, and most of the improvements are from improved "engineering" rather than improved "algorithms". In contrast, my work focuses on developing algorithms with rigorous theoretical analysis that improve the sustainability of IT. In particular, this thesis seeks to exploit the flexibilities of cloud workloads both (i) in time by scheduling delay-tolerant workloads and (ii) in space by routing requests to geographically diverse data centers. These opportunities allow data centers to adaptively respond to renewable availability, varying cooling efficiency, and fluctuating energy prices, while still meeting performance requirements. The design of the enabling algorithms is however very challenging because of limited information, non-smooth objective functions and the need for distributed control. Novel distributed algorithms are developed with theoretically provable guarantees to enable the "follow the renewables" routing. Moving from theory to practice, I helped HP design and implement industry's first Net-zero Energy Data Center.

The second thrust of this thesis is to use IT systems to improve the sustainability and efficiency of our energy infrastructure through data center demand response. The main challenges as we integrate more renewable sources to the existing power grid come from the fluctuation and unpredictability of renewable generation. Although energy storage and reserves can potentially solve the issues, they are very costly. One promising alternative is to make the cloud data centers demand responsive. The potential of such an approach is huge.

To realize this potential, we need adaptive and distributed control of cloud data centers and new electricity market designs for distributed electricity resources. My work is progressing in both directions. In particular, I have designed online algorithms with theoretically guaranteed performance for data center operators to deal with uncertainties under popular demand response programs. Based on local control rules of customers, I have further designed new pricing schemes for demand response to align the interests of customers, utility companies, and the society to improve social welfare.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.