Disturbance Propagation in Dynamical Networks

Subir Sarker

doi:10.7273/000005113

This dissertation focuses on the propagation of disturbances in dynamical networks. Specifically, we use graph-theoretic analysis to characterize the disturbance responses in sparsely actuated and sensed dynamical networks. In addition, we define the stability of disturbance propagation in general dynamical networks and investigate it in inverter-based microgrids. In general, the dissertation addresses three distinct but interconnected research problems: 1) spatial analysis of network synchronization processes, 2) disturbance propagation stability in dynamical networks, and 3) controllability assessment of dynamical networks. First, we consider the input-output analysis of a discrete-time linear network synchronization model using graph-theoretical analysis. We demonstrate that some input-output metrics, including lp gains, frequency responses, frequency-band energy, and Markov parameters, show a spatial decrescence property in which they are nonincreasing along separating cutsets away from the disturbance source. Based on these spatial results, we characterize the signal-to-noise (SNR) ratio in diffusive networks and define disturbance propagation stability in general dynamical networks. The spatial analysis is then extended to a class of nonlinear dynamical systems, the modified DeGroot-Friedkin model for opinion formation in networks. The nonlinear self-confidence dynamics in this model also show spatial degradation under certain constraints. We develop a graph-preserving model reduction algorithm with nonlinear self-confidence dynamics using these spatial results. All these formal results are illustrated through suitable examples. Second, motivated by the spatial analysis of dynamical networks, we consider disturbance propagation stability notions for a synchronization process of homogeneous subsystems coupled linearly. Here, we present a general definition of disturbance propagation stability based on the degradation of response norms with separating cutsets away from the disturbance source. Based on this definition, the network synchronization model is propagation stable if it is asymptotically internally (Lyapunov) stable, and the maximum singular values of the subsystem model's local transfer matrices over all frequencies are upper bounded by unity. The analysis can be extended to an induced subnetwork and simplified for the single-input single-output (SISO) case. Then, we consider the disturbance propagation stability analysis for an inverter-based microgrid's angle dynamics using spatial degradation of H∞ or H2 gains. We show that the propagation stability of the microgrid depends on the inverter control parameters. We also characterize the disturbance in the frequency domain if the microgrid is propagation unstable and describe resilient design techniques considering the trade-off between coherency and disturbance propagation. Finally, the results are demonstrated through simulation. Third, the controllability of an inverter-based microgrid is analyzed using graph-theoretic analysis. Specifically, we consider shaping the small-signal dynamics of a droop-controlled microgrid using an additional input. This problem corresponds to the structural controllability assessment of the microgrid. We demonstrate that the microgrid model is structurally controllable for inputs at the digraph's zero-forcing sets (a graph-theoretic property). The graph-theoretic result is verified by computing control input energies at the zero-forcing sets in two test systems.

Disturbance Propagation in Dynamical Networks

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