This dissertation examines modeling, design and control challenges associatedwith two classes of power converters: a direct current-direct current (DC-DC) step-down (buck) regulator and a 3-phase (3-ϕ) 4-wire direct current-alternating current (DC-AC) inverter. These are widely used for power transfer in a variety of industrial and personal applications. This motivates the precise quantification of conditions under which existing modeling and design methods yield satisfactory designs, and the study of alternatives when they don’t. This dissertation describes a method utilizing Fourier components of the input square wave and the inductor-capacitor (LC) filter transfer function, which doesn’t require the small ripple approximation. Then, trade-offs associated with the choice of the filter order are analyzed for integrated buck converters with a constraint on their chip area. Design specifications which would justify using a fourth or sixth order filter instead of the widely used second order one are examined. Next, sampled-data (SD) control of a buck converter is analyzed. Three methods for the digital controller design are studied: analog design followed by discretization, direct digital design of a discretized plant, and a “lifting” based method wherein the sampling time is incorporated in the design process by lifting the continuous-time design plant before doing the controller design. Specifically, controller performance is quantified by studying the induced-L2 norm of the closed loop system for a range of switching/sampling frequencies. In the final segment of this dissertation, the inner-outer control loop, employed in inverters with an inductor-capacitor-inductor (LCL) output filter, is studied. Closed loop sensitivities for the loop broken at the error and the control are examined, demonstrating that traditional methods only address these properties for one loop-breaking point. New controllers are then provided for improving both sets of properties.

Models using feature interactions have been applied successfully in many areas such as biomedical analysis, recommender systems. The popularity of using feature interactions mainly lies in (1) they are able to capture the nonlinearity of the data compared with linear effects and (2) they enjoy great interpretability. In this thesis, I propose a series of formulations using feature interactions for real world problems and develop efficient algorithms for solving them.

Specifically, I first propose to directly solve the non-convex formulation of the weak hierarchical Lasso which imposes weak hierarchy on individual features and interactions but can only be approximately solved by a convex relaxation in existing studies. I further propose to use the non-convex weak hierarchical Lasso formulation for hypothesis testing on the interaction features with hierarchical assumptions. Secondly, I propose a type of bi-linear models that take advantage of interactions of features for drug discovery problems where specific drug-drug pairs or drug-disease pairs are of interest. These models are learned by maximizing the number of positive data pairs that rank above the average score of unlabeled data pairs. Then I generalize the method to the case of using the top-ranked unlabeled data pairs for representative construction and derive an efficient algorithm for the extended formulation. Last but not least, motivated by a special form of bi-linear models, I propose a framework that enables simultaneously subgrouping data points and building specific models on the subgroups for learning on massive and heterogeneous datasets. Experiments on synthetic and real datasets are conducted to demonstrate the effectiveness or efficiency of the proposed methods.

This research mainly focuses on improving the utilization of photovoltaic (PV) re-sources in distribution systems by reducing their variability and uncertainty through the integration of distributed energy storage (DES) devices, like batteries, and smart PV in-verters. The adopted theoretical tools include statistical analysis and convex optimization. Operational issues have been widely reported in distribution systems as the penetration of PV resources has increased. Decision-making processes for determining the optimal allo-cation and scheduling of DES, and the optimal placement of smart PV inverters are con-sidered. The alternating current (AC) power flow constraints are used in these optimiza-tion models. The first two optimization problems are formulated as quadratically-constrained quadratic programming (QCQP) problems while the third problem is formu-lated as a mixed-integer QCQP (MIQCQP) problem. In order to obtain a globally opti-mum solution to these non-convex optimization problems, convex relaxation techniques are introduced. Considering that the costs of the DES are still very high, a procedure for DES sizing based on OpenDSS is proposed in this research to avoid over-sizing.

Some existing convex relaxations, e.g. the second order cone programming (SOCP) relaxation and semidefinite programming (SDP) relaxation, which have been well studied for the optimal power flow (OPF) problem work unsatisfactorily for the DES and smart inverter optimization problems. Several convex constraints that can approximate the rank-1 constraint X = xxT are introduced to construct a tighter SDP relaxation which is referred to as the enhanced SDP (ESDP) relaxation using a non-iterative computing framework. Obtaining the convex hull of the AC power flow equations is beneficial for mitigating the non-convexity of the decision-making processes in power systems, since the AC power flow constraints exist in many of these problems. The quasi-convex hull of the quadratic equalities in the AC power bus injection model (BIM) and the exact convex hull of the quadratic equality in the AC power branch flow model (BFM) are proposed respectively in this thesis. Based on the convex hull of BFM, a novel convex relaxation of the DES optimizations is proposed. The proposed approaches are tested on a real world feeder in Arizona and several benchmark IEEE radial feeders.

Advances in data collection technologies have made it cost-effective to obtain heterogeneous data from multiple data sources. Very often, the data are of very high dimension and feature selection is preferred in order to reduce noise, save computational cost and learn interpretable models. Due to the multi-modality nature of heterogeneous data, it is interesting to design efficient machine learning models that are capable of performing variable selection and feature group (data source) selection simultaneously (a.k.a bi-level selection). In this thesis, I carry out research along this direction with a particular focus on designing efficient optimization algorithms. I start with a unified bi-level learning model that contains several existing feature selection models as special cases. Then the proposed model is further extended to tackle the block-wise missing data, one of the major challenges in the diagnosis of Alzheimer's Disease (AD). Moreover, I propose a novel interpretable sparse group feature selection model that greatly facilitates the procedure of parameter tuning and model selection. Last but not least, I show that by solving the sparse group hard thresholding problem directly, the sparse group feature selection model can be further improved in terms of both algorithmic complexity and efficiency. Promising results are demonstrated in the extensive evaluation on multiple real-world data sets.

This dissertation considers an integrated approach to system design and controller design based on analyzing limits of system performance. Historically, plant design methodologies have not incorporated control relevant considerations. Such an approach could result in a system that might not meet its specifications (or one that requires a complex control architecture to do so). System and controller designers often go through several iterations in order to converge to an acceptable plant and controller design. The focus of this dissertation is on the design and control an air-breathing hypersonic vehicle using such an integrated system-control design framework. The goal is to reduce the number of system-control design iterations (by explicitly incorporate control considerations in the system design process), as well as to influence the guidance/trajectory specifications for the system. Due to the high computational costs associated with obtaining a dynamic model for each plant configuration considered, approximations to the system dynamics are used in the control design process. By formulating the control design problem using bilinear and polynomial matrix inequalities, several common control and system design constraints can be simultaneously incorporated into a vehicle design optimization. Several design problems are examined to illustrate the effectiveness of this approach (and to compare the computational burden of this methodology against more traditional approaches).

Transmission expansion planning (TEP) is a complex decision making process that requires comprehensive analysis to determine the time, location, and number of electric power transmission facilities that are needed in the future power grid. This dissertation investigates the topic of solving TEP problems for large power systems. The dissertation can be divided into two parts. The first part of this dissertation focuses on developing a more accurate network model for TEP study. First, a mixed-integer linear programming (MILP) based TEP model is proposed for solving multi-stage TEP problems. Compared with previous work, the proposed approach reduces the number of variables and constraints needed and improves the computational efficiency significantly. Second, the AC power flow model is applied to TEP models. Relaxations and reformulations are proposed to make the AC model based TEP problem solvable. Third, a convexified AC network model is proposed for TEP studies with reactive power and off-nominal bus voltage magnitudes included in the model. A MILP-based loss model and its relaxations are also investigated. The second part of this dissertation investigates the uncertainty modeling issues in the TEP problem. A two-stage stochastic TEP model is proposed and decomposition algorithms based on the L-shaped method and progressive hedging (PH) are developed to solve the stochastic model. Results indicate that the stochastic TEP model can give a more accurate estimation of the annual operating cost as compared to the deterministic TEP model which focuses only on the peak load.