Avoidance Markov Metrics and Node Pivotality Ranking

We introduce the avoidance Markov metrics and theories which provide more flexibility in the design of random walk and impose new conditions on the walk to avoid (or transit) a specific node (or a set of nodes) before the stopping criteria. These theories help with applications that cannot be modeled by classical Markov chains and require more flexibility and intricacy in their modeling. Specifically, we use them for the pivotality ranking of the nodes in a network reachabilities. More often than not, it is not sufficient simply to know whether a source node $s$ can reach a target node $t$ in the network and additional information associated with reachability, such as how long or how many possible ways node $s$ may take to reach node $t$, is required. In this paper, we analyze the pivotality of the nodes which capture how pivotal a role that a node $k$ or a subset of nodes $S$ may play in the reachability from node $s$ to node $t$ in a given network. Through some synthetic and real-world network examples, we show that these metrics build a powerful ranking tool for the nodes based on their pivotality in the reachability

Structure- & Physics- Preserving Reductions of Power Grid Models

The large size of multiscale, distribution and transmission, power grids hinder fast system-wide estimation and real-time control and optimization of operations. This paper studies graph reduction methods of power grids that are favorable for fast simulations and follow-up applications. While the classical Kron reduction has been successful in reduced order modeling of power grids with traditional, hierarchical design, the selection of reference nodes for the reduced model in a multiscale, distribution and transmission, network becomes ambiguous. In this work we extend the use of the iterative Kron reduction by utilizing the electric grid's graph topology for the selection of reference nodes, consistent with the design features of multiscale networks. Additionally, we propose further reductions by aggregation of coherent subnetworks of triangular meshes, based on the graph topology and network characteristics, in order to preserve currents and build another power-flow equivalent network. Our reductions are achieved through the use of iterative aggregation of sub-graphs that include general tree structures, lines, and triangles. Important features of our reduction algorithms include that: (i) the reductions are, either, equivalent to the Kron reduction, or otherwise produce a power-flow equivalent network; (ii) due to the former mentioned power-flow equivalence, the reduced network can model the dynamic of the swing equations for a lossless, inductive, steady state network; (iii) the algorithms efficiently utilize hash-tables to store the sequential reduction steps

Deterministic Random Walk Preconditioning for Power Grid Analysis

Iterative linear equation solvers depend on high quality preconditioners to achieve fast convergence. For sparse symmetric systems arising from large power grid analysis problems, however, preconditioners generated by traditional incomplete Cholesky factorizations are usually of low quality, resulting in slow convergence. On the other hand, although preconditioners generated by random walks are quite effective to reduce the number of iterations, it takes considerable amount of time to compute them in a stochastic manner. In this paper, we propose a new preconditioning technique for power grid analysis, named deterministic random walk, that combines the advantages of the above two approaches. Our proposed algorithm computes the preconditioners in a deterministic manner to reduce computation time, while achieving similar quality as stochastic random walk preconditioning by modifying fill-ins to compensate dropped entries. We have proved that for such compensation scheme, our algorithm will not fail for certain matrix orderings, which otherwise cannot be guaranteed by traditional incomplete factorizations. We demonstrate that by incorporating our proposed preconditioner, a conjugate gradient solver is able to outperform a state-of-the-art algebraic multigrid preconditioned solver on public IBM power grid benchmarks for DC power grid analysis, while potentially remaining very efficient for transient analysis. 1