Recent work on the internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes. Such deletions include, for example, the failure of internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process, but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.

This paper discusses actively involving highly distributed loads in power system control actions; an overview of system control objectives is provided.By Duncan S. Callaway, Member IEEE, and Ian A. Hiskens, Fellow IEEE ABSTRACT | This paper discusses conceptual frameworks for actively involving highly distributed loads in power system control actions. The context for load control is established by providing an overview of system control objectives, including economic dispatch, automatic generation control, and spinning reserve. The paper then reviews existing initiatives that seek to develop load control programs for the provision of power system services. We then discuss some of the challenges to achieving a load control scheme that balances device-level objectives with power system-level objectives. One of the central premises of the paper is that, in order to achieve full responsiveness, direct load control (as opposed to price response) is required to enable fast time scale, predictable control opportunities, especially for the provision of ancillary services such as regulation and contingency reserves. Centralized, hierarchical, and distributed control architectures are discussed along with benefits and disadvantages, especially in relation to integration with the legacy power system control architecture. Implications for the supporting communications infrastructure are also considered. Fully responsive load control is illustrated in the context of thermostatically controlled loads and plug-in electric vehicles.

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