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The power distribution grid is typically unobservable due to a lack of measurements. While deploying more sensors can alleviate this issue, it also presents new challenges related to data aggregation and the underlying communication infrastructure. Therefore, developing state estimation methods that enhance situational awareness at the grid edge with compressed measurements is critical. For this purpose, a suite of sparsity-based approaches that exploit the correlation among states/measurements in spatial as well as temporal domains have been proposed recently. This paper presents a systematic comparison and evaluation of these approaches. Specifically, the performance and complexity of spatial methods (1-D compressive sensing and matrix completion) and spatio-temporal methods (2-D compressive sensing and tensor completion) are compared using the IEEE 37 and IEEE 123 bus test systems. Additionally, new robust formulations of these sparsity-based methods are derived and shown to be robust to bad data and network parameter uncertainties. Among the sparsity-based approaches, compressive sensing methods tend to outperform matrix completion and tensor completion methods in terms of error performance.

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