Exact topology reconstruction of radial dynamical systems with applications to distribution system of the power grid

Abstract

In this article we present a method to reconstruct the interconnectedness of dynamically related stochastic processes, where the interactions are bi-directional and the underlying topology is a tree. Our approach is based on multivariate Wiener filtering which recovers spurious edges apart from the true edges in the topology reconstruction. The main contribution of this work is to show that all spurious links obtained using Wiener filtering can be eliminated if the underlying topology is a tree based on which we present a three stage network reconstruction procedure for trees. We illustrate the effectiveness of the method developed by applying it on a typical distribution system of the electric grid. I. INTRODUCTIONNetworks underpin a powerful framework for modeling and analysis of large scale dynamical systems. Applications include neuroscience [1], financial markets [2], protein dynamics [3], climate sciences [4] and the power grid [5]. Moreover, networks play an indispensable role in building foundational aspects of control theory [6], statistical inference [7] and optimization theory [8]. The compactness of representation and the capability of unveiling influences, cause-effect relationships and dependencies amongst many variables are some of the key attributes enabled by network based approaches [9], [10]. An essential aspect of many studies is to determine a graphical representation of how multiple sub-systems/agents interact from measured time series data. It is often the case that active manipulation of the system is prohibited or not possible; for example, in financial markets the prices of stocks are available as data but it is not possible (or not allowed) to manipulate the prices. In many cases, the influences between sub-systems/ agents is mutual, thus separating source and destination or cause and effect in such cases is not meaningful.In this article, we are concerned with the task of unveiling the network topology that relates multiple linear dynamical systems from temporal data, where it is not possible to excite the system externally. Here, we assume that the underlying network is bi-directed, that is, the influences between agents is mutual and describing cause-effect relationships is not obvious. We further restrict the study to systems where the interaction flow is well characterized by a tree structure.

Physics informed topology learning in networks of linear dynamical systems

“…For radial topologies (undirected connected graph with no cycles) associated with bi-directed generative graphs, it is possible to distinguish between true edges between neighbors and spurious two hop neighbor edges in G M by using a local graph separation rule as presented in [19]. However for bi-directed generative graphs with topologies having cycles or loops, such graph separation results do not hold in general [31].…”

“…For consensus dynamics, a decentralized and distributed topology learning scheme is presented in [17] and [18] respectively. The works summarized above, primarily use a mix of signal processing or optimization schemes or structural restrictions like radial topology [19] to infer the topology. A crucial characteristic is that, none of the above works, utilize any knowledge about the underlying physics of the system toward topology learning.…”

Graphical Models in Meshed Distribution Grids: Topology Estimation, Change Detection & Limitations

“…Example of such schemes include greedy methods [5], [6], voltage signature based methods [7], [8], probing schemes [9], imposing graph cycle constraints [10] and iterative schemes for addressing missing data [11], [12]. In contrast to the referred work that employ static voltage samples, learning schemes that exploit dynamic voltage measurements are reported in [13], [14].…”

An MILP Approach for Distribution Grid Topology Identification using Inverter Probing

“…Several works build on the properties of second-order statistics from smart meter data to infer feeder topologies [1], [2], [3]. A Wiener filtering approach using wide-sense stationary processes on radial networks is put forth in [4]. Nonetheless, sample statistics converge to their ensemble values only after a large number of grid data has been collected, thus rendering topology estimates possibly obsolete.…”