With the advent of distributed energy resources (DERs), electric vehicles, and demand-response programs, grid operators are in dire need of new monitoring and design tools that help improve efficiency, reliability, and stability of modern power networks. To this end, the work in this thesis explores a generalized modeling and analysis framework for two pertinent tasks: i) learning loads via grid probing, and; ii) optimizing power grid topologies for stability. Distribution grids currently lack comprehensive real-time metering. Nevertheless, grid operators require precise knowledge of loads and renewable generation to accomplish any feeder optimization task. At the same time, new grid technologies, such as solar panels and energy storage units are interfaced via inverters with advanced sensing and actuation capabilities. In this context, we first put forth the idea of engaging power electronics to probe an electric grid and record its voltage response at actuated and metered buses to infer non-metered loads. Probing can be accomplished by commanding inverters to momentarily perturb their power injections. Multiple probing actions can be induced within a few tens of seconds. Load inference via grid probing is formulated as an implicit nonlinear system identification task, which is shown to be topologically observable under certain conditions. The analysis holds for single- and multi-phase grids, radial or meshed, and applies to phasor or magnitude-only voltage data. Using probing to learn non-constant-power loads is also analyzed as a special case. Once a probing setup is deemed topologically observable, a methodology for designing probing injections abiding by inverter and network constraints to improve load estimates is provided. The probing task under noisy phasor and non-phasor data is tackled using a semidefinite-program relaxation. As a second contribution, we also study the effect of topology on the linear time-invariant dynamics of power networks. For a variety of stability metrics, a unified framework based on the H2-norm of the system is presented. The proposed framework assesses the robustness of power grids to small disturbances and is used to study the optimal placement of new lines on existing networks as well as the design of radial topologies for new networks. / Doctor of Philosophy / Increased penetration of distributed energy resources such as solar panels, wind farms, and energy storage systems is forcing utilities to rethink how they design and operate their power networks. To ensure efficient and reliable operation of distribution networks and to perform any grid-wide optimization or dispatch tasks, the system operator needs to precisely know the net load (energy output) of every customer. However, due to the sheer extent of distribution networks (millions of customers) and low investment interest in the past, distribution grids have limited metering infrastructure. Nevertheless, data from grid sensors comprised of voltage and load measurements are readily available from a subset of customers at high temporal resolution. In addition, the smart inverters found in solar panels, energy storage units, and electric vehicles can be controlled within microseconds. The work in this thesis explores how the proliferation of grid sensors together with the controllability of smart inverters can be leveraged for inferring the non-metered loads i.e., energy output of customers that are not equipped with smart inverters/sensors. In addition to the load learning task, this thesis also presents a modeling and analysis framework to study the optimal design of topologies (how customers are electrically inter-connected) for improving stability of our power networks.

<p>The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors.</p><p>In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor.</p><p>These methods are also implemented in MATLAB.</p><p>We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.</p>

The question about how to determine the rank of a tensor has been widely studied in the literature. However the analytical methods to compute the decomposition of tensors have not been so much developed even for low-rank tensors. In this report we present analytical methods for finding real and complex PARAFAC decompositions of 2 x 2 x 2 tensors before computing the actual rank of the tensor. These methods are also implemented in MATLAB. We also consider the question of how best lower-rank approximation gives rise to problems of degeneracy, and give some analytical explanations for these issues.

This master's thesis addresses numerical methods of computing the typical ranks of tensors over the real numbers and explores some properties of tensors over finite fields. We present three numerical methods to compute typical tensor rank. Two of these have already been published and can be used to calculate the lowest typical ranks of tensors and an approximate percentage of how many tensors have the lowest typical ranks (for some tensor formats), respectively. The third method was developed by the authors with the intent to be able to discern if there is more than one typical rank. Some results from the method are presented but are inconclusive. In the area of tensors over nite filds some new results are shown, namely that there are eight GLq(2) GLq(2) GLq(2)-orbits of 2 2 2 tensors over any finite field and that some tensors over Fq have lower rank when considered as tensors over Fq2 . Furthermore, it is shown that some symmetric tensors over F2 do not have a symmetric rank and that there are tensors over some other finite fields which have a larger symmetric rank than rank.