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Controlling the Self-organizing Dynamics in Sandpile Models by Failure Tolerance and Applications to Economic and Ecological Systems
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Grasshopper population dynamics are an important part of the North American rangeland ecosystem and an important factor in the economies that derive from the rangeland. Outbreak dynamics have plagued management strategies in the rangeland, and attempts to find simple, linear and mechanistic solutions to both understanding and predicting the dynamics have proved fruitless. These efforts to ground theory in a correspondence with the "real" world, including whether the population dynamics are ultimately density dependent or density independent, have generated abundant heat but little light. We suggest that a pragmatic approach, in which theories are taken to be "tools" rather than competing claims of truth, has greater promise to move ecological research in a constructive direction. Two recent non-linear approaches exploiting the tools of complexity science provide insights relevant to explaining and forecasting population dynamics. Observation and data collection were used to structure models derived from catastrophe theory and self-organized criticality. These models indicate that nonlinear processes are important in the dynamics of the outbreaks. And the conceptual structures of these approaches provide clear, albeit constrained or contingent, implications for pest managers.
We show that, although these two frameworks, catastrophe theory and self-organized criticality, are very different, the frequency distributions of time series from both systems result in power law relationships. Further, we show that a simple lattice-based model, similar to SOC but structured on the biology of the grasshoppers gives a spatial time series similar to data over a 50-year span and the frequency distribution is also a power law relationship. This demonstration exemplifies how a "both-and" rather than an "either-or" approach to ecological modeling, in which the useful elements of particular theories or conceptual structures are extracted, may provide a way forward in addressing particularly difficult ecological problems.
In this paper the interactions between component failures are quantified and the interaction matrix and interaction network are obtained. The quantified interactions can capture the general propagation patterns of the cascades from utilities or simulation, thus helping to better understand how cascading failures propagate and to identify key links and key components that are crucial for cascading failure propagation. By utilizing these interactions a high-level probabilistic model called interaction model is proposed to study the influence of interactions on cascading failure risk and to support online decision-making. It is much more time efficient to first quantify the interactions between component failures with fewer original cascades from a more detailed cascading failure model and then perform the interaction model simulation than it is to directly simulate a large number of cascades with a more detailed model. Interaction-based mitigation measures are suggested to mitigate cascading failure risk by weakening key links, which can be achieved in real systems by wide area protection such as blocking of some specific protective relays. The proposed interaction quantifying method and interaction model are validated with line outage data generated by the AC OPA cascading simulations on the IEEE 118-bus system.
In this paper, the multi-type branching process is applied to describe the statistics of line outages, the load shed, and isolated buses. The offspring mean matrix of the multi-type branching process is estimated by the Expectation Maximization (EM) algorithm and can quantify the extent of outage propagation. The joint distribution of two types of outages is estimated by the multi-type branching process via the Lagrange-Good inversion. The proposed model is tested with data generated by the AC OPA cascading simulations on the IEEE 118-bus system. The largest eigenvalues of the offspring mean matrix indicate that the system is closer to criticality when considering the interdependence of different types of outages. Compared with empirically estimating the joint distribution of the total outages, good estimate is obtained by using the multi-type branching process with a much smaller number of cascades, thus greatly improving the efficiency. It is shown that the multi-type branching process can effectively predict the distribution of the load shed and isolated buses even when there are no data of them.
Branching processes can be applied to simulated cascading data to describe the statistics of the cascades and quickly predict the distribution of blackout sizes. We improve the procedures for discretizing load shed data so that a Galton-Watson branching process may be applied. The branching process parameters such as average propagation are estimated from simulated cascades and the branching process is then used to estimate the distribution of blackout size. We test the estimated distributions with line outage and load shed data generated by the improved OPA and AC OPA cascading simulations on the IEEE 118-bus system and the Northeast Power Grid of China.
In this paper a blackout model that considers the slow process at the beginning of blackouts is proposed based on the improved ORNL-PSerc-Alaska (OPA) model. It contains two layers of iteration. The inner iteration, which describes the fast dynamics of the system, simulates the power system cascading failure, including the tree contact and failure of lines caused by heating. The simulation of protective relays and the dispatching center is also improved to make it closer to practical conditions. The outer iteration, which describes the long-term slow dynamics of the system, adds the simulation of tree growth and utility vegetation management (UVM). Typical blackout process with slow process is simulated on the Northeast Power Grid of China and self-organized criticality (SOC) characteristicof this system is analyzed with the
proposed model. The effectiveness of the proposed model is verified by the simulation results.
Controlling self-organizing systems is challenging because the system responds to the controller. Here, we develop a model that captures the essential self-organizing mechanisms of Bak-Tang-Wiesenfeld (BTW) sandpiles on networks, a self-organized critical (SOC) system. This model enables studying a simple control scheme that determines the frequency of cascades and that shapes systemic risk. We show that optimal strategies exist for generic cost functions and that controlling a subcritical system may drive it to criticality. This approach could enable controlling other self-organizing systems.
A control scheme to reduce the size of avalanches of the Bak-Tang-Wiesenfeld
model on complex networks is proposed. Three network types are considered:
those proposed by Erd\H{o}s-Renyi, Goh-Kahng-Kim, and a real network
representing the main connections of the electrical power grid of the western
United States. The control scheme is based on the idea of triggering avalanches
in the highest degree nodes that are near to become critical. We show that this
strategy works in the sense that the dissipation of mass occurs most locally
avoiding larger avalanches. We also compare this strategy with a random
strategy where the nodes are chosen randomly. Although the random control has
some ability to reduce the probability of large avalanches, its performance is
much worse than the one based on the choice of the highest degree nodes.
Finally, we argue that the ability of the proposed control scheme is related to
its ability to reduce the concentration of mass on the network.