State estimation is an important tool for continuously monitoring the power system and aims to recover the underlying system voltage phasors, given supervisory control and data acquisition (SCADA) measurements and a model that encompasses the system topology and specifications. To ensure an accurate state estimation, it is essential to have the capability of detecting bad data. Assuming that the network parameters are known and the measurement devices are correctly calibrated, the main source of bad data is topological errors in the model. In this dissertation, we propose a methodology for robust power system state estimation (PSSE) modeled by AC power flow equations when there exists a small number of topological errors. The developed technique utilizes the availability of a large number of SCADA measurements and minimizes the L-1 norm of nonconvex residuals augmented by a nonlinear, but convex, regularizer. Representing the power network by a graph, we first study the properties of the solution obtained from the proposed NLAV estimator and demonstrate that, under mild conditions, this solution identifies a small subgraph of the network that contains the topological errors in the model used for the state estimation problem. Then, we introduce a method that can efficiently detect the topological errors by searching over the identified subgraph. In addition, we develop a theoretical upper bound on the state estimation error to guarantee the accuracy of the proposed state estimation technique.

The power flow equations are nonlinear, and may admit multiple solutions. In the past, the conventional wisdom was to assume that the solution becomes unique by restricting it to "realistic'' or "physically realizable'' values. However, various examples in the literature show that multiple solutions may persist even after restricting either voltage magnitudes or phase angle differences to "physically realizable'' values. This dissertation establishes sufficient conditions for the uniqueness of AC power flow solutions via the monotonic relationship between real power flow and the phase angle difference. More specifically, we prove that the P-Theta power flow problem has at most one solution for any acyclic or GSP graph. In addition, for arbitrary cyclic power networks, we show that multiple distinct solutions cannot exist under the assumption that angle differences across the lines are bounded by some limit related to the maximal girth of the network. We also introduce a series-parallel operator and show that this operator obtains a reduced and easier-to-analyze model for the power system without changing the uniqueness of power flow solutions.

In the next part of this dissertation, the above work is extended and we establish general necessary and sufficient conditions for the uniqueness of P-Theta power flow solutions in an AC power system using properties of the monotone regime and the power network topology. We show that the necessary and sufficient conditions can lead to tighter sufficient conditions for the uniqueness in several special cases. Our results are based on the previously introduced notion of maximal girth and a new notion of maximal eye. Moreover, we develop a series-parallel reduction method and search-based algorithms for computing the maximal eye and maximal girth, which are necessary for the uniqueness analysis. Reduction to a single line using the proposed reduction method is guaranteed for 2-vertex-connected Series-Parallel graphs.

In the final part of this dissertation, we present a methodology based on homotopy to find the globally optimal solutions of nonconvex optimization problems. Optimal power flow (OPF) is a fundamental problem in power systems analysis for determining the steady-state operating point of a power network that minimizes the generation cost. In anticipation of component failures, such as transmission line or generator outages, it is also important to find optimal corrective actions for the power flow distribution over the network. The problem of finding these post-contingency solutions to the OPF problem is challenging due to the nonconvexity of the power flow equations and the large number of contingency cases in practice. In this paper, we introduce a homotopy method to solve for the post-contingency actions, which involves a series of intermediate optimization problems that gradually transform the original OPF problem into each contingency-OPF problem. We show that given a global solution to the original OPF problem, a global solution to the contingency problem can be obtained using this homotopy method, under some conditions. With simulations on Polish and other European networks, we demonstrate that the effectiveness of the proposed homotopy method is dependent on the choice of the homotopy path and that homotopy yields an improved solution in many cases.

Monoglycerides, glycerol monoesters of fatty acids, offer several advantages over organic acids by increasing stability, reducing unpleasant odors, and enabling the gradual release of bioactive substances (i.e., organic fatty acids) throughout the intestine, where they can positively affect intestinal integrity. These properties make monoglycerides attractive nutritional interventions for improving animal health and growth. In this context, three studies were performed to investigate the potential beneficial effects of monoglycerides as a practical strategy to improve disease resistance and overall health in pigs.The first study aimed to evaluate the potential of a monoglyceride blend and zinc glycinate as sustainable methods for improving intestinal mucosal integrity in weaned pigs. In vitro cell culture models (porcine enterocyte cell line [IPEC-J2] and porcine alveolar macrophages [PAM]) were used to assess intestinal barrier and immunomodulatory functions. IPEC-J2 cells treated with 250 and 1,000 μg/mL of monoglycerides had increased (P < 0.05) transepithelial electrical resistance (TEER) values at 48 and 72 h after treatment, while 5 μg/mL of zinc glycinate showed increased (P < 0.0001) TEER values only at 72 h after treatment, compared with the control group. Lipopolysaccharide (LPS) challenge increased (P < 0.05) the production of tumor necrosis factor-alpha (TNF-α) and interleukin-1 beta (IL-1β) from PAM. In the non-challenge group, 50 or 100 μg/mL of monoglycerides stimulated (P < 0.05) TNF-α and IL-1β production from PAM. Treatment with 25 or 100 μg/mL of zinc glycinate also enhanced (P < 0.05) TNF-α production from PAM. In LPS-treated PAM, 1,000 μg/mL of monoglycerides increased (P < 0.05) IL-1β production, while zinc glycinate suppressed (P < 0.0001) the secretion of TNF-α and IL-1β at the doses of 100, 250, and 500 μg/mL. The results from in vitro culture assays indicate that monoglycerides have a beneficial effect on strengthening epithelial barrier function, while zinc glycinate may have strong anti-inflammatory benefits. The second study evaluated the effects of dietary monoglycerides (blend of short- and medium-chain fatty acids; BalanGut™ LS L; BASF SE) on the gut health and immunity of weaned pigs experimentally infected with an enterotoxigenic Escherichia coli (ETEC) F18. Pigs in high-dose zinc oxide (ZNO; 3,000 mg/kg) and antibiotic groups had lower (P < 0.05) severity of diarrhea than control, but the severity of diarrhea was not different between antibiotic and monoglycerides groups. Pigs fed with monoglycerides or ZNO had lower (P < 0.05) serum haptoglobin on d 2 or 5 post-inoculation (PI) than control. Pigs in ZNO had greater (P < 0.05) goblet cell numbers per villus, villus area and height, and villus height-to-crypt depth ratio (VH:CD) in duodenum on d 5 PI than pigs in control, monoglycerides, and antibiotic groups. Pigs supplemented with monoglycerides, ZNO, or antibiotic had reduced (P < 0.05) ileal crypt depth compared with control on d 5 PI, contributing to the increase (P = 0.06) in VH:CD. Consistently, pigs in ZNO expressed the lowest (P < 0.05) TNFa, IL6, IL10, IL12, IL1A, IL1B, and PTGS2 in ileal mucosa on d 5 PI, while no difference was observed in the expression of those genes between ZNO and monoglycerides. Supplementation of ZNO or antibiotic had significant impacts on metabolic pathways in the serum compared with control, particularly on carbohydrate and amino acids metabolism, while limited impacts on serum metabolites were observed in monoglycerides group when compared with control. Pigs supplemented with ZNO had greater (P < 0.05) growth performance than other treatments, but no difference was observed in average daily feed intake between ZNO and monoglycerides groups during the post-challenge period. The second study indicates that supplementation of monoglyceride blend may enhance disease resistance of weaned pigs by alleviating the severity of diarrhea and mitigating intestinal and systemic inflammation, although the effectiveness may not be comparable to high-dose zinc oxide. The third study was conducted to explore the effects of dietary supplementation of organic acids (blend of formic acid, lactic acid, and sodium formate; Acitra G20C, Eastman), monoglycerides (blend of short- and medium-chain fatty acids; Entero-Nova 410C, Eastman), and combination of both acid-based additives on diarrhea, immune responses, and growth performance of ETEC F18-challenged weaned pigs. Supplementation of organic acids, monoglycerides, or their combination significantly reduced (P < 0.05) the frequency of diarrhea compared with control group. ETEC F18 infection increased (P < 0.05) counts of total white blood cells, neutrophils, and lymphocytes on d 5 and 14 PI, compared with d 0. Supplementation of the combination of organic acids and monoglycerides reduced (P < 0.05) the counts of total white blood cells, lymphocytes, and the ratio of neutrophils to lymphocytes, and tended (P < 0.10) to reduce neutrophil count on d 5 PI, compared with control. Pigs fed with monoglycerides had higher (P < 0.05) neutrophil counts than pigs in the control group on d 14 PI. Pigs fed with monoglycerides tended (P < 0.10) to have lower serum granulocyte-macrophage colony-stimulating factor, interleukin-1 alpha, and TNF-α concentrations on d 0, 5, or 14 PI compared with control. However, supplementation of organic acids, monoglycerides, or the combination of both had limited impacts on growth performance throughout the experiment. The third study suggests that dietary supplementation of organic acid blend, monoglyceride blend, and or their combination may alleviate diarrhea, intestinal damage, or inflammation caused by ETEC infection in pigs. Moreover, the potential for synergistic effects related to immune-modulatory effect observed in the combination group requires further investigation. In summary, monoglycerides investigated in the present studies have positive effects on disease resistance and the overall health of weaned pigs under ETEC F18 infection, by improving gut integrity and modulating local and systemic inflammation.