Paradigm and paradox in power networks
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Well known in the theory of network flows, Braess paradox states that adding path(s) to a congested road network may increase overall journey time. In transportation networks, the phenomenon results from selfish routing. In power systems, an analogous increase in congestion can arise as a consequence of Kirchhoff's laws, suggesting opportunities to optimize grid topology. The thesis starts with the discussion of Braess-like congestion phenomena in linear circuits. We prove that adding electrical path(s) always increases congestion in networks powered by voltage sources, while the opposite in networks driven by current sources. Although such predictability is not present in networks controlled by a mixture of voltage and current sources, our results offer a clean decomposition that completely separates the effect of current sources and voltage sources on total loss. The culmination of this research is a set of four equivalent methods of computing I^2R loss in mixed-source networks. We go on to explore network decomposition in combination with greedy sequential line switching heuristics to address the NP-hardness of power grid topology control. By means of some low order examples, it is shown that within a reasonably large class of greedy heuristics, none can be found that perform better than the others across all grid topologies. Despite this cautionary tale, statistical evidence indicates that, among three most representative heuristics, the global greedy heuristic is most computationally intensive but has the best chance of reducing generation cost while enforcing connectivity. The final part of the thesis presents a new approach to grid decomposition using vertex cut sets. We show that each vertex cut set and corresponding grid decomposition establishes a natural upper bound on the interactions between subgrids as nodal injections are regulated within each. Using such decomposition, it becomes possible to isolate congestion effects to a relatively small subgrid. A fast grid decomposition heuristic based on vertex cut sets and locational marginal prices is then proposed and studied through simulations on IEEE 118-bus system. On average, the computational cost is significantly reduced and the generation cost saving is similar to what is obtained with a global greedy algorithm.