Supermodal Decomposition of the Linear Swing Equation for Multilayer Networks

Abstract

We study the swing equation in the case of a multilayer network in which generators and motors are modeled differently; namely, the model for each generator is given by second order dynamics and the model for each motor is given by first order dynamics. We also remove the commonly used assumption of equal damping coefficients in the second order dynamics. Under these general conditions, we are able to obtain a decomposition of the linear swing equation into independent modes describing the propagation of small perturbations. In the process, we identify symmetries affecting the structure and dynamics of the multilayer network and derive an essential model based on a 'quotient network'. We then compare the dynamics of the full network and that of the quotient network and obtain a modal decomposition of the error dynamics. We also provide a method to quantify the steady-state error and the maximum overshoot error. Two case studies are presented to illustrate application of our method.

Exact Decomposition of Optimal Control Problems via Simultaneous Block Diagonalization of Matrices

“…Given a general networked system represented by the graph G, and system dynamics (23), one can follow the 4 steps below to decouple the system into multiple OCPs and solve them in parallel to reduce the computation time.…”

“…Symmetries are common in both natural and engineering systems [18], [19], [20], [21], [22], [23]. The analysis of symmetries has been exploited in various disciplines such as semi-definite programming [24], [25], [26], network synchronization [27], [28], arithmetic optimization methods [29], the backward computation of reachable sets for nonlinear discrete-time control systems [30], and stability of interconnected subsystems [31], thus making it a popular approach for reducing the computation complexity associated with high dimensional problems.…”