Alejandro D. Dominguez-Garcia

Nitin H. Vaidya, Christoforos N. Hadjicostis, Alejandro D. Dominguez-Garcia

In this two-part paper, we consider multicomponent systems in which each component can iteratively exchange information with other components in its neighborhood in order to compute, in a distributed fashion, the average of the components' initial values or some other quantity of interest (i.e., some function of these initial values). In particular, we study an iterative algorithm for computing the average of the initial values of the nodes. In this algorithm, each component maintains two sets of variables that are updated via two identical linear iterations. The average of the initial values of the nodes can be asymptotically computed by each node as the ratio of two of the variables it maintains. In the first part of this paper, we show how the update rules for the two sets of variables can be enhanced so that the algorithm becomes tolerant to communication links that may drop packets, independently among them and independently between different transmission times. In this second part, by rewriting the collective dynamics of both iterations, we show that the resulting system is mathematically equivalent to a finite inhomogenous Markov chain whose transition matrix takes one of finitely many values at each step. Then, by using e a coefficients of ergodicity approach, a method commonly used for convergence analysis of Markov chains, we prove convergence of the robustified consensus scheme. The analysis suggests that similar convergence should hold under more general conditions as well.

Alejandro D. Dominguez-Garcia, Christoforos N. Hadjicostis, Nitin H. Vaidya

This two-part paper discusses robustification methodologies for linear-iterative distributed algorithms for consensus and coordination problems in multicomponent systems, in which unreliable communication links may drop packets. We consider a setup where communication links between components can be asymmetric (i.e., component j might be able to send information to component i, but not necessarily vice-versa), so that the information exchange between components in the system is in general described by a directed graph that is assumed to be strongly connected. In the absence of communication link failures, each component i maintains two auxiliary variables and updates each of their values to be a linear combination of their corresponding previous values and the corresponding previous values of neighboring components (i.e., components that send information to node i). By appropriately initializing these two (decoupled) iterations, the system components can asymptotically calculate variables of interest in a distributed fashion; in particular, the average of the initial conditions can be calculated as a function that involves the ratio of these two auxiliary variables. The focus of this paper to robustify this double-iteration algorithm against communication link failures. We achieve this by modifying the double-iteration algorithm (by introducing some additional auxiliary variables) and prove that the modified double-iteration converges almost surely to average consensus. In the first part of the paper, we study the first and second moments of the two iterations, and use them to establish convergence, and illustrate the performance of the algorithm with several numerical examples. In the second part, in order to establish the convergence of the algorithm, we use coefficients of ergodicity commonly used in analyzing inhomogeneous Markov chains.

Madi Zholbaryssov, Alejandro D. Dominguez-Garcia

In this paper, we consider the problem of optimally coordinating the response of a group of distributed energy resources (DERs) in distribution systems by solving the so-called optimal power flow (OPF) problem. The OPF problem is concerned with determining an optimal operating point, at which some cost function, e.g., generation cost or power losses, is minimized, and operational constraints are satisfied. To solve the OPF problem, we propose distributed algorithms that are able to operate over time-varying communication networks and have geometric convergence rate. We solve the second-order cone program (SOCP) relaxation of the OPF problem for radial distribution systems, which is formulated using the so-called DistFlow model. Theoretical results are further supported by the numerical simulations.