Optimal measurement allocation for parametric model identification of electrical networks

Abstract

In this paper we develop optimal sensor siting methods along the edges of a large network of electrical oscillators to identify a parametric model for the network using dynamic measurements of electrical signals corrupted with Gaussian noise. We pose the identification problem as estimation of four essential parameters for each edge in the network, namely the real and imaginary components of the edge-weight, or equivalently, the resistance and reactance of the tie-line connecting any two oscillators, and the inertias of the oscillators connected by this edge. We then formulate the Cramer-Rao bounds for the estimates of these four unknown parameters using three fundamental outputs - namely, the magnitude, the phase angle and the frequency of the voltage phasor along each edge, and show that the bounds are functions of the sensor locations on the edges as well as of the contribution of each variable in the combined output. We finally state the condition for finding the optimal sensor location and the optimal signal combination to achieve the tightest Cramer-Rao bound. The problem is first addressed for open-loop networks and, thereafter, for networks where outputs measured at desired locations on the edges are fed back to the nodes to improve transient performance. We show that unlike the first case where the open-loop configuration allows us to optimize the bounds in a distributed fashion for each individual edge, for the latter situation the problem no longer has a decoupled structure under the influence of feedback, and must be carried out in a centralized fashion.