Optimal uncertainty quantification via convex optimization and relaxation

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

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.

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.

Real-time load-side control of electric power systems

Two trends are emerging from modern electric power systems: the growth of renewable (e.g., solar and wind) generation, and the integration of information technologies and advanced power electronics. The former introduces large, rapid, and random fluctuations in power supply, demand, frequency, and voltage, which become a major challenge for real-time operation of power systems. The latter creates a tremendous number of controllable intelligent endpoints such as smart buildings and appliances, electric vehicles, energy storage devices, and power electronic devices that can sense, compute, communicate, and actuate. Most of these endpoints are distributed on the load side of power systems, in contrast to traditional control resources such as centralized bulk generators. This thesis focuses on controlling power systems in real time, using these load side resources. Specifically, it studies two problems.

(1) Distributed load-side frequency control: We establish a mathematical framework to design distributed frequency control algorithms for flexible electric loads. In this framework, we formulate a category of optimization problems, called optimal load control (OLC), to incorporate the goals of frequency control, such as balancing power supply and demand, restoring frequency to its nominal value, restoring inter-area power flows, etc., in a way that minimizes total disutility for the loads to participate in frequency control by deviating from their nominal power usage. By exploiting distributed algorithms to solve OLC and analyzing convergence of these algorithms, we design distributed load-side controllers and prove stability of closed-loop power systems governed by these controllers. This general framework is adapted and applied to different types of power systems described by different models, or to achieve different levels of control goals under different operation scenarios. We first consider a dynamically coherent power system which can be equivalently modeled with a single synchronous machine. We then extend our framework to a multi-machine power network, where we consider primary and secondary frequency controls, linear and nonlinear power flow models, and the interactions between generator dynamics and load control.

(2) Two-timescale voltage control: The voltage of a power distribution system must be maintained closely around its nominal value in real time, even in the presence of highly volatile power supply or demand. For this purpose, we jointly control two types of reactive power sources: a capacitor operating at a slow timescale, and a power electronic device, such as a smart inverter or a D-STATCOM, operating at a fast timescale. Their control actions are solved from optimal power flow problems at two timescales. Specifically, the slow-timescale problem is a chance-constrained optimization, which minimizes power loss and regulates the voltage at the current time instant while limiting the probability of future voltage violations due to stochastic changes in power supply or demand. This control framework forms the basis of an optimal sizing problem, which determines the installation capacities of the control devices by minimizing the sum of power loss and capital cost. We develop computationally efficient heuristics to solve the optimal sizing problem and implement real-time control. Numerical experiments show that the proposed sizing and control schemes significantly improve the reliability of voltage control with a moderate increase in cost.

Sustainable energy solutions for irrigation and harvesting in developing countries

One of the critical problems currently being faced by agriculture industry in developing nations is the alarming rate of groundwater depletion. Irrigation accounts for over 70% of the total groundwater withdrawn everyday. Compounding this issue is the use of polluting diesel generators to pump groundwater for irrigation. This has made irrigation not only the biggest consumer of groundwater but also one of the major contributors to green house gases. The aim of this thesis is to present a solution to the energy-water nexus. To make agriculture less dependent on fossil fuels, the use of a solar-powered Stirling engine as the power generator for on-farm energy needs is discussed. The Stirling cycle is revisited and practical and ideal Stirling cycles are compared. Based on agricultural needs and financial constraints faced by farmers in developing countries, the use of a Fresnel lens as a solar-concentrator and a Beta-type Stirling engine unit is suggested for sustainable power generation on the farms. To reduce the groundwater consumption and to make irrigation more sustainable, the conceptual idea of using a Stirling engine in drip irrigation is presented. To tackle the shortage of over 37 million tonnes of cold-storage in India, the idea of cost-effective solar-powered on-farm cold storage unit is discussed.

Effects of self energy of the ions on the double layer structure and properties at the dielectric interface

Although numerous theoretical efforts have been put forth, a systematic, unified and predictive theoretical framework that is able to capture all the essential physics of the interfacial behaviors of ions, such as the Hofmeister series effect, Jones-Ray effect and the salt effect on the bubble coalescence remain an outstanding challenge. The most common approach to treating electrostatic interactions in the presence of salt ions is the Poisson-Boltzmann (PB) theory. However, there are many systems for which the PB theory fails to offer even a qualitative explanation of the behavior, especially for ions distributed in the vicinity of an interface with dielectric contrast between the two media (like the water-vapor/oil interface). A key factor missing in the PB theory is the self energy of the ion.

In this thesis, we develop a self-consistent theory that treats the electrostatic self energy (including both the short-range Born solvation energy and the long-range image charge interactions), the nonelectrostatic contribution of the self energy, the ion-ion correlation and the screening effect systematically in a single framework. By assuming a finite charge spread of the ion instead of using the point-charge model, the self energy obtained by our theory is free of the divergence problems and gives a continuous self energy across the interface. This continuous feature allows ions on the water side and the vapor/oil side of the interface to be treated in a unified framework. The theory involves a minimum set of parameters of the ion, such as the valency, radius, polarizability of the ions, and the dielectric constants of the medium, that are both intrinsic and readily available. The general theory is first applied to study the thermodynamic property of the bulk electrolyte solution, which shows good agreement with the experiment result for predicting the activity coefficient and osmotic coefficient.

Next, we address the effect of local Born solvation energy on the bulk thermodynamics and interfacial properties of electrolyte solution mixtures. We show that difference in the solvation energy between the cations and anions naturally gives rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The miscibility of the mixture can either increases or decreases depending on the competition between the solvation energy and translation entropy of the ions. The interfacial tension shows a non-monotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations, and decreases approximately as the square root of the salt concentration for dilute solutions, which is in agreement with the Jones-Ray effect observed in experiment.

Next, we investigate the image effects on the double layer structure and interfacial properties near a single charged plate. We show that the image charge repulsion creates a depletion boundary layer that cannot be captured by a regular perturbation approach. The correct weak-coupling theory must include the self-energy of the ion due to the image charge interaction. The image force qualitatively alters the double layer structure and properties, and gives rise to many non-PB effects, such as nonmonotonic dependence of the surface energy on concentration and charge inversion. The image charge effect is then studied for electrolyte solutions between two plates. For two neutral plates, we show that depletion of the salt ions by the image charge repulsion results in short-range attractive and long-range repulsive forces. If cations and anions are of different valency, the asymmetric depletion leads to the formation of an induced electrical double layer. For two charged plates, the competition between the surface charge and the image charge effect can give rise to like- charge attraction.

Then, we study the inhomogeneous screening effect near the dielectric interface due to the anisotropic and nonuniform ion distribution. We show that the double layer structure and interfacial properties is drastically affected by the inhomogeneous screening if the bulk Debye screening length is comparable or smaller than the Bjerrum length. The width of the depletion layer is characterized by the Bjerrum length, independent of the salt concentration. We predict that the negative adsorption of ions at the interface increases linearly with the salt concentration, which cannot be captured by either the bulk screening approximation or the WKB approximation. For asymmetric salt, the inhomogeneous screening enhances the charge separation in the induced double layer and significantly increases the value of the surface potential.

Finally, to account for the ion specificity, we study the self energy of a single ion across the dielectric interface. The ion is considered to be polarizable: its charge distribution can be self-adjusted to the local dielectric environment to minimize the self energy. Using intrinsic parameters of the ions, such as the valency, radius, and polarizability, we predict the specific ion effect on the interfacial affinity of halogen anions at the water/air interface, and the strong adsorption of hydrophobic ions at the water/oil interface, in agreement with experiments and atomistic simulations.

The theory developed in this work represents the most systematic theoretical technique for weak-coupling electrolytes. We expect the theory to be more useful for studying a wide range of structural and dynamic properties in physicochemical, colloidal, soft-matter and biophysical systems.

I. The rate and mechanism of the vapor-phase reaction between water and parts-per-million concentrations of nitrogen dioxide. II. The rate and mechanism of the air oxidation of parts-per-million concentrations of nitric oxide in the presence of water vapor.

Part I

A study of the thermal reaction of water vapor and parts-per-million concentrations of nitrogen dioxide was carried out at ambient temperature and at atmospheric pressure. Nitric oxide and nitric acid vapor were the principal products. The initial rate of disappearance of nitrogen dioxide was first order with respect to water vapor and second order with respect to nitrogen dioxide. An initial third-order rate constant of 5.5 (± 0.29) x 10⁴ liter² mole^-2 sec^-1 was found at 25˚C. The rate of reaction decreased with increasing temperature. In the temperature range of 25˚C to 50˚C, an activation energy of -978 (± 20) calories was found.

The reaction did not go to completion. From measurements as the reaction approached equilibrium, the free energy of nitric acid vapor was calculated. This value was -18.58 (± 0.04) kilocalories at 25˚C.

The initial rate of reaction was unaffected by the presence of oxygen and was retarded by the presence of nitric oxide. There were no appreciable effects due to the surface of the reactor. Nitric oxide and nitrogen dioxide were monitored by gas chromatography during the reaction.

Part II

The air oxidation of nitric oxide, and the oxidation of nitric oxide in the presence of water vapor, were studied in a glass reactor at ambient temperatures and at atmospheric pressure. The concentration of nitric oxide was less than 100 parts-per-million. The concentration of nitrogen dioxide was monitored by gas chromatography during the reaction.

For the dry oxidation, the third-order rate constant was 1.46 (± 0.03) x 10⁴ liter² mole^-2 sec^-1 at 25˚C. The activation energy, obtained from measurements between 25˚C and 50˚C, was -1.197 (±0.02) kilocalories.

The presence of water vapor during the oxidation caused the formation of nitrous acid vapor when nitric oxide, nitrogen dioxide and water vapor combined. By measuring the difference between the concentrations of nitrogen dioxide during the wet and dry oxidations, the rate of formation of nitrous acid vapor was found. The third-order rate constant for the formation of nitrous acid vapor was equal to 1.5 (± 0.5) x 10⁵ liter² mole^-2 sec^-1 at 40˚C. The reaction rate did not change measurably when the temperature was increased to 50˚C. The formation of nitric acid vapor was prevented by keeping the concentration of nitrogen dioxide low.

Surface effects were appreciable for the wet tests. Below 35˚C, the rate of appearance of nitrogen dioxide increased with increasing surface. Above 40˚C, the effect of surface was small.

Part I: Latent heat of vaporization of n-decane. Part II: Heat and mass transfer from a cylinder placed in a turbulent air stream - Sherwood and Frössling numbers as a function of free stream turbulence level and Reynolds number

Part I

The latent heat of vaporization of n-decane is measured calorimetrically at temperatures between 160° and 340°F. The internal energy change upon vaporization, and the specific volume of the vapor at its dew point are calculated from these data and are included in this work. The measurements are in excellent agreement with available data at 77° and also at 345°F, and are presented in graphical and tabular form.

Part II

Simultaneous material and energy transport from a one-inch adiabatic porous cylinder is studied as a function of free stream Reynolds Number and turbulence level. Experimental data is presented for Reynolds Numbers between 1600 and 15,000 based on the cylinder diameter, and for apparent turbulence levels between 1.3 and 25.0 per cent. n-heptane and n-octane are the evaporating fluids used in this investigation.

Gross Sherwood Numbers are calculated from the data and are in substantial agreement with existing correlations of the results of other workers. The Sherwood Numbers, characterizing mass transfer rates, increase approximately as the 0.55 power of the Reynolds Number. At a free stream Reynolds Number of 3700 the Sherwood Number showed a 40% increase as the apparent turbulence level of the free stream was raised from 1.3 to 25 per cent.

Within the uncertainties involved in the diffusion coefficients used for n-heptane and n-octane, the Sherwood Numbers are comparable for both materials. A dimensionless Frössling Number is computed which characterizes either heat or mass transfer rates for cylinders on a comparable basis. The calculated Frössling Numbers based on mass transfer measurements are in substantial agreement with Frössling Numbers calculated from the data of other workers in heat transfer.