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Center for Nonlinear Studies |
We discuss the modeling and synchronization problem for structure-preserving power system models with either frequency-dependent or linear load models. The latter load model leads to the network-reduced model of the generator swing dynamics. We exploit the relationship between the considered power network models and the well-known Kuramoto model of coupled oscillators. Extending methods from transient stability analysis, synchronization theory, and consensus protocols, we establish static synchronization conditions for the dynamic power network and coupled-oscillator models. First, we focus on a network of coupled first-order Kuramoto oscillators and derive purely algebraic conditions for synchronization. Our conditions are necessary and sufficient for a complete and homogeneous network, they are sufficient for a topological network with heterogeneous coupling, and they improve upon previously-available tests for the Kuramoto model. Second, we discuss the extension of these synchronization conditions to the second-order coupled-oscillator models arising in power networks. This extension from first-order to second-order dynamics can be made rigorous by means of topological conjugacy arguments, by a singular perturbation analysis, or by strict-mechanical Lyapunov functions. In the end, we are able to state concise and purely algebraic conditions that relate synchronization in a power network to the underlying network state, parameters, and topology. Third, we analyze the network-reduction process relating the network-reduced and the more detailed structure-preserving power system model. The network reduction process, termed Kron reduction, is characterized by iterative Schur complementation of the admittance matrix. A detailed algebraic and graph-theoretic analysis of the Kron reduction process allows us to extend the synchronization conditions obtained for the network-reduced model to the structure-preserving model. In the end, we are able to state one spectral and one resistance-based condition for synchronization. Time permitting, we briefly touch upon other networked-control approaches to power network problems.
Host: Misha Chertkov, chertkov@lanl.gov, 665-8119