Deepjyoti Deka

Estimating distribution grid topologies: A graphical learning based approach

Distribution grids represent the final tier in electric networks consisting of medium and low voltage lines that connect the distribution substations to the end-users/loads. Traditionally, distribution networks have been operated in a radial topology that may be changed from time to time. Due to absence of a significant number of real-time line monitoring devices in the distribution grid, estimation of the topology/structure is a problem critical for its observability and control. This paper develops a novel graphical learning based approach to estimate the radial operational grid structure using voltage measurements collected from the grid loads. The learning algorithm is based on conditional independence tests for continuous variables over chordal graphs and has wide applicability. It is proven that the scheme can be used for several power flow laws (DC or AC approximations) and more importantly is independent of the specific probability distribution controlling individual buss power usage. The complexity of the algorithm is discussed and its performance is demonstrated by simulations on distribution test cases.

Structure Learning in Power Distribution Networks

Traditional power distribution networks suffer from a lack of real-time observability. This complicates development and implementation of new smart-grid technologies, such as those related to demand response, outage detection and management, and improved load monitoring. In this paper, inspired by proliferation of metering technology, we discuss topology estimation problems in structurally loopy but operationally radial distribution grids from measurements, for example, voltage data, which are either already available or can be made available with a relatively minor investment. The primary objective of this paper is to learn the operational layout of the grid. Further, the structure learning algorithm is extended to cases with missing data, where available observations are limited to a fraction of the grid nodes. The algorithms are computationally efficient—polynomial in time—which is proven theoretically and illustrated in numerical experiments on a number of test cases. The techniques developed can be applied to detect line failures in real time as well as to understand the scope of possible adversarial attacks on the grid.

Learning topology of the power distribution grid with and without missing data

Distribution grids refer to the part of the power grid that delivers electricity from substations to the loads. Structurally a distribution grid is operated in one of several radial/tree-like topologies that are derived from an original loopy grid graph by opening switches on some lines. Due to limited presence of real-time switch monitoring devices, the operating structure needs to be estimated indirectly. This paper presents a new learning algorithm that uses only nodal voltage measurements to determine the operational radial structure. The algorithm is based on the key result stating that the correct operating structure is the optimal solution of the minimum-weight spanning tree problem over the original loopy graph where weights on all permissible edges/lines (open or closed) is the variance of nodal voltage difference at the edge ends. Compared to existing work, this spanning tree based approach has significantly lower complexity as it does not require information on line parameters. Further, a modified learning algorithm is developed for cases when the input voltage measurements are limited to only a subset of the total grid nodes. Performance of the algorithms (with and without missing data) is demonstrated by experiments on test cases.

Learning topology of distribution grids using only terminal node measurements

Distribution grids include medium and low voltage lines that are involved in the delivery of electricity from substation to end-users/loads. A distribution grid is operated in a radial/tree-like structure, determined by switching on or off lines from an underling loopy graph. Due to the presence of limited real-time measurements, the critical problem of fast estimation of the radial grid structure is not straightforward. This paper presents a new learning algorithm that uses measurements only at the terminal or leaf nodes in the distribution grid to estimate its radial structure. The algorithm is based on results involving voltages of node triplets that arise due to the radial structure. The polynomial computational complexity of the algorithm is presented along with a detailed analysis of its working. The most significant contribution of the approach is that it is able to learn the structure in certain cases where available measurements are confined to only half of the nodes. This represents learning under minimum permissible observability. Performance of the proposed approach in learning structure is demonstrated by experiments on test radial distribution grids.

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

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.

Ensemble Control of Cycling Energy Loads: Markov Decision Approach

A Markov decision process (MDP) framework is adopted to represent ensemble control of devices with cyclic energy consumption patterns, e.g., thermostatically controlled loads. Specifically we utilize and develop the class of MDP models previously coined linearly solvable MDPs, that describe optimal dynamics of the probability distribution of an ensemble of many cycling devices. Two principally different settings are discussed. First, we consider optimal strategy of the ensemble aggregator balancing between minimization of the cost of operations and minimization of the ensemble welfare penalty, where the latter is represented as a KL-divergence between actual and normal probability distributions of the ensemble. Then, second, we shift to the demand response setting modeling the aggregator’s task to minimize the welfare penalty under the condition that the aggregated consumption matches the targeted time-varying consumption requested by the system operator. We discuss a modification of both settings aimed at encouraging or constraining the transitions between different states. The dynamic programming feature of the resulting modified MDPs is always preserved; however, “linear solvability” is lost fully or partially, depending on the type of modification. We also conducted some (limited in scope) numerical experimentation using the formulations of the first setting. We conclude by discussing future generalizations and applications.

Optimal topology design for disturbance minimization in power grids

The transient response of power grids to external disturbances influences their stable operation. This paper studies the effect of topology in linear time-invariant dynamics of different power grids. For a variety of objective functions, a unified framework based on H2 norm is presented to analyze the robustness to ambient fluctuations. Such objectives include loss reduction, weighted consensus of phase angle deviations, oscillations in nodal frequency, and other graphical metrics. The framework is then used to study the problem of optimal topology design for robust control goals of different grids. For radial grids, the problem is shown as equivalent to the hard “optimum communication spanning tree” problem in graph theory and a combinatorial topology construction is presented with bounded approximation gap. Extended to loopy (meshed) grids, a greedy topology design algorithm is discussed. The performance of the topology design algorithms under multiple control objectives are presented on both loopy and radial test grids. Overall, this paper analyzes topology design algorithms on a broad class of control problems in power grid by exploring their combinatorial and graphical properties.

Tractable structure learning in radial physical flow networks

Physical Flow Networks are different infrastructure networks that allow the flow of physical commodities through edges between its constituent nodes. These include power grid, natural gas transmission network, water pipelines etc. In such networks, the flow on each edge is characterized by a function of the nodal potentials on either side of the edge. Further the net flow in and out of each node is conserved. Learning the structure and state of physical networks is necessary for optimal control as well as to quantify its privacy needs. We consider radial flow networks and study the problem of learning the operational network from a loopy graph of candidate edges using statistics of nodal potentials. Based on the monotonic properties of the flow functions, the key result in this paper shows that if variance of the difference of nodal potentials is used to weight candidate edges, the operational edges form the minimum spanning tree in the loopy graph. Under realistic conditions on the statistics of nodal injection (consumption or production), we provide a greedy structure learning algorithm with quasilinear computational complexity in the number of candidate edges in the network. Our learning framework is very general due to two significant attributes. First it is independent of the specific marginal distributions of nodal potentials and only uses order properties in their second moments. Second, the learning algorithm is agnostic to exact flow functions that relate edge flows to corresponding potential differences and is applicable for a broad class of networks with monotonic flow functions. We demonstrate the efficacy of our work through realistic simulations on diverse physical flow networks and discuss possible extensions of our work to other regimes.

Learning Exact Topology of a Loopy Power Grid from Ambient Dynamics

Estimation of the operational topology of the power grid is necessary for optimal market settlement and reliable dynamic operation of the grid. This paper presents a novel framework for topology estimation for general power grids (loopy or radial) using time-series measurements of nodal voltage phase angles that arise from the swing dynamics. Our learning framework utilizes multivariate Wiener filtering to unravel the interaction between fluctuations in voltage angles at different nodes and identifies operational edges by considering the phase response of the elements of the multivariate Wiener filter. The performance of our learning framework is demonstrated through simulations on standard IEEE test cases.

Topology Learning in Radial Distribution Grids

Overview Chapter Accurate estimation of the state and topology of the distribution grid is hindered by the limited placement of real-time flow meters and breaker statuses at distribution grid lines. In recent years, increasing presence of smart devices and sensors at households have made measurements of consumption and voltages available at distribution buses. This chapter discusses greedy algorithms to learn the grid topology using voltage measurements collected at a subset of the buses in the distribution grid. The distribution grids are operated in a radial topology. This topological restriction leads to provable trends in voltage second moments (covariances) and enables the design of our learning algorithms. For the case where voltage measurements are available at all grid buses, our framework does not require any additional information related to line impedances of grid lines or consumption statistics at buses to estimate the operational topology. Further in presence of such information, we demonstrate guaranteed topology learning in scenarios with varying fraction of “missing” buses that have no voltage measurements. The efficiency of the algorithms is highlighted by their computational complexity that scales polynomially in the number of grid buses.