## Performance Measures for Oscillator Networks

##### Abstract

Oscillator networks consist of a set of simple subsystems, e.g. damped harmonic oscillators that interact with each other across a network with a specified structure. Such networks of coupled oscillators serve as a model for many systems such as power grids, vehicle platoons, and biological networks. Even though the dynamics of each oscillator are simple, the coupling between them can produce complex behavior. One possible behavior is synchronization, where all of the oscillators reach a state where their relative phase angles are constant and their frequencies are uniform. This work examines the synchronization performance of oscillator networks, i.e. how well the network maintains synchrony in the face of persistent disturbances. Specifically, we define a class of performance measures for oscillator networks as the H2-norm of particular input-output linear systems. This class of performance measures corresponds to measuring the average value of a quadratic form of the oscillator phases when stochastic disturbances are applied to some subset of the oscillators. Depending on the specific quadratic form that is chosen, this performance measure can correspond to a variety of physically meaningful and domain specific quantities. For example, it can be used to quantify the total interactions between oscillators during resynchronization after a disturbance. This quantity corresponds to the transient resistive losses in maintaining synchronous operation in a power network. Alternatively, one can instead measure the network coherence, which quantifies how closely the oscillator network acts like a single rigid body. Our results demonstrate a strong connection between the concept of effective resistance and our class of performance measures. For example, our results make precise the intuitive notion that more "tightly connected'' oscillator networks are more coherent by showing that the maximum effective resistance in the network is the correct notion of connectivity. We consider applications of the work to both power grids and vehicle platoons with local and absolute (global) velocity feedback. For power grids we use our effective resistance based results to obtain novel bounds on the resistive losses due to generators maintaining synchrony. For vehicle platoons we investigate the coherence in the platoon as a performance measure. We show that for large scale platoons local velocity feedback performs worse than absolute velocity feedback under certain conditions related to the asymptotic behavior of the maximum effective resistance in the underlying graph.