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In order to explain previously observed deviations of measured plasma velocities from velocities predicted by computer solutions based on a snowplow model, the experimental and computer data are here extended over a much wider range of electrode configurations, capacitor potentials, and gas densities in the shock tube. At low densities the experimental velocity curves fall below the predicted curves because of material evaporated from insulation and electrodes. At high gas densities the experimental velocity curve for the largest center electrode coincides with the predicted curve, but for center electrodes of successively smaller diameter t~e measured curves lie progressively farther above, but parallel to, those predicted, up to a factor of 5.5 times fo~ the smalle~t electrode use~. Th~se results are explainable in terms of a modified snowplow model postulatmg an effective annular regIOn adjacent to the center electrode, whose thickness is of the same order as the electrode radius, such that the force due to the magnetic pressure inside this annulus acts on only the gas contained within it to produce the measured velocity.

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.