High-resolution distribution-level phasor measurement units (D-PMUs), a.k.a, micro-PMU (µPMU), have recently been deployed in the power distribution grids in California and elsewhere. They frequently measure and report voltage and current phasors. Although synchrophasor sensors entail an unprecedented level of visibility over the power grid, working with extremely large data sets that they generate is still a challenge. Besides, it is cost-prohibitive to deploy a large number of synchrophasor sensors on a single distribution feeder. In this thesis, we focus on the analysis of the event signatures that exist in synchrophasor measurements to overcome the challenges for optimal usage of synchrophasor data.The informative parts of synchrophasor data are associated with the changes occurring in the electricity grid during its operation. These changes in the system states are called "events". Thus far, relatively few research efforts have been made towards understanding and utilizing of the events in distribution synchrophasors which carry information about the distribution feeder, its equipment, and various loads. Our goal in this thesis is to develop new methodologies to unmask innovative use-cases and applications which use only a few and locationally scarce D-PMU measurements that capture frequent and locationally abundant events occurring in distribution network. In this regard, we introduce the differential phasor representation of an event and propose an analytical framework to make use of differential phasors. A new class of fundamental methodologies is developed that involves constructing the equivalent circuit for the power distribution network in the differential mode which is generated based on the compensation theorem in circuit theory. Subsequently, we propose several practical applications: localizing events and identifying the root-cause of the events, developing event-based tracking state estimation for the purpose of real-time monitoring, identifying line segments and estimating line parameters, and detecting and locating high impedance faults. Furthermore, several applications are developed based on the synchrophasor data obtained from normal operation of the distribution system, including, estimating line switches status and identifying distribution system topology, and developing a linear state estimation involving synchrophasor data along with conventional power distribution system measurements
