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There is an increasing need to apply rigorous model-order-reduction techniques in the analysis of large-scale networks of inverter-based distributed generation resources due to the limitations of existing simulation tools. Various coherency-based aggregation techniques have long been used to construct reduced-order dynamic models of large-scale synchronous machine (SM) networks. Such techniques have the advantage of preserving the nonlinear nature of the dynamic model throughout the order-reduction process, enabling the efficient and accurate analysis of large-scale network dynamics during large disturbances such as fault events. This paper proposes the application of a rigorous coherency-based aggregation technique to the analysis of large-scale networks of grid-forming droop-controlled inverters. A rapid and powerful generalized eigenvalue perturbation technique for coherency identification, previously only applied to SM networks, is adapted to grid-forming droop-controlled inverter networks. The resulting reduced-order models are physically insightful and are capable of accurately reproducing the system response in the aftermath of large disturbances. For some networks, a rigorously-derived condition of coherency can be difficult to achieve, given the expected range of L–C–L filter impedances. To remedy this limitation, the potential for high-bandwidth inverter control to enforce the conditions that allow for coherency of droop-controlled inverters has been investigated and confirmed using a controller hardware-in-the loop testbed. Using this approach, the use of simple nonlinear aggregate inverter models to accurately model large sections of the inverter network can be more rigorously justified.

We present a hierarchical optimization architecture for large-scale power networks that overcomes limitations of fully centralized and fully decentralized architectures. The architecture leverages principles of multigrid computing schemes, which are widely used in the solution of partial differential equations on massively parallel computers. The top layer of the architecture uses a coarse representation of the entire network while the bottom layer is composed of a family of decentralized optimization agents each operating on a network subdomain at full resolution. We use an alternating direction method of multipliers (ADMM) framework to drive coordination of the decentralized agents. We show that state and dual information obtained from the top layer can be used to accelerate the coordination of the decentralized optimization agents and to recover optimality for the entire system. We demonstrate that the hierarchical architecture can be used to manage large collections of microgrids.

Advances in sensor technologies provide a method to accurately assess activity levels of people with stroke in their community. This information could be used to determine the effectiveness of rehabilitation interventions as well as provide behavior-enhancing feedback. The purpose of this study was to assess the accuracy of a novel shoe-based sensor system (SmartShoe) to identify different functional postures and steps in people with stroke. The SmartShoe system consists of five force-sensitive resistors built into a flexible insole and an accelerometer on the back of the shoe. Pressure and acceleration data are sent via Bluetooth to a smart phone.Participants with stroke wore the SmartShoe while they performed activities of daily living (ADLs) in sitting, standing, and walking positions. Data from four participants were used to develop a multilayer perceptron artificial neural network (ANN) to identify sitting, standing, and walking. A signal-processing algorithm used data from the pressure sensors to estimate the number of steps taken while walking. The accuracy, precision, and recall of the ANN for identifying the three functional postures were calculated with data from a different set of participants. Agreement between steps identified by SmartShoe and actual steps taken was analyzed by the Bland Altman method.The SmartShoe was able to accurately identify sitting, standing, and walking. Accuracy, precision, and recall were all greater than 95%. The mean difference between steps identified by SmartShoe and actual steps was less than one step.The SmartShoe was able to accurately identify different functional postures, using a unique combination of pressure and acceleration data, of people with stroke as they performed different ADLs. There was a strong level of agreement between actual steps taken and steps identified by the SmartShoe. Further study is needed to determine whether the SmartShoe could be used to provide valid information on activity levels of people with stroke while they go about their daily lives in their home and community.

Pythagorean sets have interested mathematicians for over 2000 years. They may be of interest to some students in the mathematics classes of the senior high school and the college.