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Abstract-This paper studies the problem of controlling complex networks, that is, the joint problem of selecting a set of control nodes and of designing a control input to steer a network to a target state. For this problem (i) we propose a metric to quantify the difficulty of the control problem as a function of the required control energy, (ii) we derive bounds based on the system dynamics (network topology and weights) to characterize the tradeoff between the control energy and the number of control nodes, and (iii) we propose an open-loop control strategy with performance guarantees. In our strategy we select control nodes by relying on network partitioning, and we design the control input by leveraging optimal and distributed control techniques. Our findings show several control limitations and properties. For instance, for Schur stable and symmetric networks: (i) if the number of control nodes is constant, then the control energy increases exponentially with the number of network nodes, (ii) if the number of control nodes is a fixed fraction of the network nodes, then certain networks can be controlled with constant energy independently of the network dimension, and (iii) clustered networks may be easier to control because, for sufficiently many control nodes, the control energy depends only on the controllability properties of the clusters and on their coupling strength. We validate our results with examples from power networks, social networks, and epidemics spreading.

We consider the problem of deriving an explicit approximate solution of the nonlinear power equations that describe a balanced power distribution network. We give sufficient conditions for the existence of a practical solution to the power flow equations, and we propose an approximation that is linear in the active and reactive power demands of the PQ buses. For this approximation, which is valid for generic power line impedances and grid topology, we derive a bound on the approximation error as a function of the grid parameters. We illustrate the quality of the approximation via simulations, we show how it can also model the presence of voltage controlled (PV) buses, and we discuss how it generalizes the DC power flow model to lossy networks.Index Terms-Power systems modeling, load flow analysis, power distribution networks, fixed point theorem.

In this paper, we consider the problem of estimating the state of a dynamical system from distributed noisy measurements. Each agent constructs a local estimate based on its own measurements and estimates from its neighbors. Estimation is performed via a two stage strategy, the first being a Kalman-like measurement update which does not require communication, and the second being an estimate fusion using a consensus matrix. In particular we study the interaction between the consensus matrix, the number of messages exchanged per sampling time, and the Kalman gain. We prove that optimizing the consensus matrix for fastest convergence and using the centralized optimal gain is not necessarily the optimal strategy if the number of exchanged messages per sampling time is small. Moreover, we showed that although the joint optimization of the consensus matrix and the Kalman gain is in general a non-convex problem, it is possible to compute them under some important scenarios. We also provide some numerical examples to clarify some of the analytical results and compare them with alternative estimation strategies.

Abstract-Suppose we have a directed graph G with set of nodes V = {1, . . . , N } and a measure x i for every node i ∈ V . The average consensus problem consists in computing the average x A = N −1 P i x i in an iterative way, exchanging information among nodes exclusively along the available edges in G. This problem appears in a number of different contexts since the 80's (decentralized computation, load balancing, clock syncronization) and, recently, has attracted much attention for possible applications to sensor networks (data fusion problems) and to coordinated control for mobile autonomous agents. Several algorithms for average consensus can be found in the literature: they differentiate on the basis of the amount of communication and computation they use, on their scalability with respect to the number of nodes, on their adaptability to time-varying graphs, and, finally, they can be deterministic or random. In this presentation we will focus on random algorithms: we will review some algorithms present in the literature and we will propose some new ones. We will present some performance results which will allow to make some comparison. Finally, we will establish some probabilistic concentration results which will give a stronger significance to previous results.

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We consider the problem of optimal reactive power compensation for the minimization of power distribution losses in a smart microgrid. We first propose an approximate model for the power distribution network, which allows us to cast the problem into the class of convex quadratic, linearly constrained, optimization problems. We then consider the specific problem of commanding the microgenerators connected to the microgrid, in order to achieve the optimal injection of reactive power. For this task, we design a randomized, gossip-like optimization algorithm. We show how a distributed approach is possible, where microgenerators need to have only a partial knowledge of the problem parameters and of the state, and can perform only local measurements. For the proposed algorithm, we provide conditions for convergence together with an analytic characterization of the convergence speed. The analysis shows that, in radial networks, the best performance can be achieved when we command cooperation among units that are neighbors in the electric topology. Numerical simulations are included to validate the proposed model and to confirm the analytic results about the performance of the proposed algorithm.

This work presents a contribution to the solution of the average agreement problem on a network with quantized links. Starting from the well-known linear diffusion algorithm, we propose a simple and effective adaptation that is able to preserve the average of states and to drive the system near to the consensus value, when a uniform quantization is applied to communication between agents. The properties of this algorithm are investigated both by a worst-case analysis and by a probabilistic analysis, and are shown to depend on the spectral properties of the evolution matrix. A special attention is devoted to the issue of the dependence of the performance on the number of agents, and several examples are given