arXiv subjects

Andrey Bernstein

Publications and source records attributed to Andrey Bernstein.

65 records · Page 4Linked to original sources

arXiv2018

Asynchronous and Distributed Tracking of Time-Varying Fixed Points

This paper develops an algorithmic framework for tracking fixed points of time-varying contraction mappings. Analytical results for the tracking error are established for the cases where: (i) the underlying contraction self-map changes at each step of the algorithm; (ii) only an imperfect information of the map is available; and, (iii) the algorithm is implemented in a distributed fashion, with communication delays and packet drops leading to asynchronous algorithmic updates. The analytical results are applicable to several classes of problems, including time-varying contraction mappings emerging from online and asynchronous implementations of gradient-based methods for time-varying convex programs. In this domain, the proposed framework can also capture the operating principles of feedback-based online algorithms, where the online gradient steps are suitably modified to accommodate actionable feedback from an underlying physical or logical network. Examples of applications and illustrative numerical results are provided.

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arXiv2018

Online Optimization as a Feedback Controller: Stability and Tracking

This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time-invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time-varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that, under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. The paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power re-balancing and tracking of DC optimal power flow points.

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arXiv2018

Load-Flow in Multiphase Distribution Networks: Existence, Uniqueness, Non-Singularity and Linear Models

This paper considers unbalanced multiphase distribution systems with generic topology and different load models, and extends the Z-bus iterative load-flow algorithm based on a fixed-point interpretation of the AC load-flow equations. Explicit conditions for existence and uniqueness of load-flow solutions are presented. These conditions also guarantee convergence of the load-flow algorithm to the unique solution. The proposed methodology is applicable to generic systems featuring (i) wye connections; (ii) ungrounded delta connections; (iii) a combination of wye-connected and delta-connected sources/loads; and, (iv) a combination of line-to-line and line-to-grounded-neutral devices at the secondary of distribution transformers. Further, a sufficient condition for the non-singularity of the load-flow Jacobian is proposed. Finally, linear load-flow models are derived, and their approximation accuracy is analyzed. Theoretical results are corroborated through experiments on IEEE test feeders.

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arXiv2017

Network-Cognizant Voltage Droop Control for Distribution Grids

This paper examines distribution systems with a high integration of distributed energy resources (DERs) and addresses the design of local control methods for real-time voltage regulation. Particularly, the paper focuses on proportional control strategies where the active and reactive output-powers of DERs are adjusted in response to (and proportionally to) local changes in voltage levels. The design of the voltage-active power and voltage-reactive power characteristics leverages suitable linear approximations of the AC power-flow equations and is network-cognizant; that is, the coefficients of the controllers embed information on the location of the DERs and forecasted non-controllable loads/injections and, consequently, on the effect of DER power adjustments on the overall voltage profile. A robust approach is pursued to cope with uncertainty in the forecasted non-controllable loads/power injections. Stability of the proposed local controllers is analytically assessed and numerically corroborated.

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arXiv2017

Bi-Level Online Control without Regret

This paper considers a bi-level discrete-time control framework with real-time constraints, consisting of several local controllers and a central controller. The objective is to bridge the gap between the online convex optimization and real-time control literature by proposing an online control algorithm with small dynamic regret, which is a natural performance criterion in nonstationary environments related to real-time control problems. We illustrate how the proposed algorithm can be applied to real-time control of power setpoints in an electrical grid.

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arXiv2016

Real-Time Minimization of Average Error in the Presence of Uncertainty and Convexification of Feasible Sets

We consider a two-level discrete-time control framework with real-time constraints where a central controller issues setpoints to be implemented by local controllers. The local controllers implement the setpoints with some approximation and advertize a prediction of their constraints to the central controller. The local controllers might not be able to implement the setpoint exactly, due to prediction errors or because the central controller convexifies the problem for tractability. In this paper, we propose to compensate for these mismatches at the level of the local controller by using a variant of the error diffusion algorithm. We give conditions under which the minimal (convex) invariant set for the accumulated-error dynamics is bounded, and give a computational method to construct this set. This can be used to compute a bound on the accumulated error and hence establish convergence of the average error to zero. We illustrate the approach in the context of real-time control of electrical grids.

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arXiv2016

Explicit Conditions on Existence and Uniqueness of Load-Flow Solutions in Distribution Networks

We present explicit sufficient conditions that guarantee the existence and uniqueness of the feasible load-flow solution for distribution networks with a generic topology (radial or meshed) modeled with positive sequence equivalents. In the problem, we also account for the presence of shunt elements. The conditions have low computational complexity and thus can be efficiently verified in a real system. Once the conditions are satisfied, the unique load-flow solution can be reached by a given fixed point iteration method of approximately linear complexity. Therefore, the proposed approach is of particular interest for modern active distribution network (ADN) setup in the context of real-time control. The theory has been confirmed through numerical experiments.

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arXiv2015

Design of Resource Agents with Guaranteed Tracking Properties for Real-Time Control of Electrical Grids

We target the problem of controlling electrical microgrids with little inertia in real time. We consider a central controller and a number of resources, where each resource is either a load, a generator, or a combination thereof, like a battery. The controller periodically computes power setpoints for the resources based on the estimated state of the grid and an overall objective, and subject to safety constraints. Each resource is augmented with a resource agent that a) implements the setpoint requests sent by the controller on the resource, and b) translates device-specific information about the resource into a device-independent representation and transmits this to the controller. We focus on the resource agents and their impact on the overall system's behavior. Intuitively, for the system to converge to the objective, the resource agents should be obedient to the requests from the controller, in the sense that the actually implemented setpoint should be close to the requested setpoint, at least on average. This can be important especially when a controller that performs continuous optimization is used (for the sake of performance) to control discrete resources (which have a discrete set of implementable setpoints). We formalize obedience by defining the notion of $c$-bounded accumulated-error. We then demonstrate its usefulness, by presenting theoretical results (for a simple scenario) and simulation results (for a more realistic setting) that indicate that, if all resource agents in the system have bounded accumulated-error, the closed-loop system converges on average to the objective. Finally, we show how to design resource agents that provably have bounded accumulated-error for various types of resources, such as resources with uncertainty (e.g., PV panels) and resources with a discrete set of implementable setpoints (e.g., on-off heating systems).

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arXiv2015

A Composable Method for Real-Time Control of Active Distribution Networks with Explicit Power Setpoints

The conventional approach for the control of distribution networks, in the presence of active generation and/or controllable loads and storage, involves a combination of both frequency and voltage regulation at different time scales. With the increased penetration of stochastic resources, distributed generation and demand response, this approach shows severe limitations in both the optimal and feasible operation of these networks, as well as in the aggregation of the network resources for upper-layer power systems. An alternative approach is to directly control the targeted grid by defining explicit and real-time setpoints for active/reactive power absorptions/injections defined by a solution of a specific optimization problem; but this quickly becomes intractable when systems get large or diverse. In this paper, we address this problem and propose a method for the explicit control of the grid status, based on a common abstract model characterized by the main property of being composable. That is to say, subsystems can be aggregated into virtual devices that hide their internal complexity. Thus the proposed method can easily cope with systems of any size or complexity. The framework is presented in this Part I, whilst in Part II we illustrate its application to a CIGRÉ low voltage benchmark microgrid. In particular, we provide implementation examples with respect to typical devices connected to distribution networks and evaluate of the performance and benefits of the proposed control framework.

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arXiv2013

Response-Based Approachability and its Application to Generalized No-Regret Algorithms

Approachability theory, introduced by Blackwell (1956), provides fundamental results on repeated games with vector-valued payoffs, and has been usefully applied since in the theory of learning in games and to learning algorithms in the online adversarial setup. Given a repeated game with vector payoffs, a target set $S$ is approachable by a certain player (the agent) if he can ensure that the average payoff vector converges to that set no matter what his adversary opponent does. Blackwell provided two equivalent sets of conditions for a convex set to be approachable. The first (primary) condition is a geometric separation condition, while the second (dual) condition requires that the set be {\em non-excludable}, namely that for every mixed action of the opponent there exists a mixed action of the agent (a {\em response}) such that the resulting payoff vector belongs to $S$. Existing approachability algorithms rely on the primal condition and essentially require to compute at each stage a projection direction from a given point to $S$. In this paper, we introduce an approachability algorithm that relies on Blackwell's {\em dual} condition. Thus, rather than projection, the algorithm relies on computation of the response to a certain action of the opponent at each stage. The utility of the proposed algorithm is demonstrated by applying it to certain generalizations of the classical regret minimization problem, which include regret minimization with side constraints and regret minimization for global cost functions. In these problems, computation of the required projections is generally complex but a response is readily obtainable.

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arXiv2012

Power Grid Vulnerability to Geographically Correlated Failures - Analysis and Control Implications

We consider power line outages in the transmission system of the power grid, and specifically those caused by a natural disaster or a large scale physical attack. In the transmission system, an outage of a line may lead to overload on other lines, thereby eventually leading to their outage. While such cascading failures have been studied before, our focus is on cascading failures that follow an outage of several lines in the same geographical area. We provide an analytical model of such failures, investigate the model's properties, and show that it differs from other models used to analyze cascades in the power grid (e.g., epidemic/percolation-based models). We then show how to identify the most vulnerable locations in the grid and perform extensive numerical experiments with real grid data to investigate the various effects of geographically correlated outages and the resulting cascades. These results allow us to gain insights into the relationships between various parameters and performance metrics, such as the size of the original event, the final number of connected components, and the fraction of demand (load) satisfied after the cascade. In particular, we focus on the timing and nature of optimal control actions used to reduce the impact of a cascade, in real time. We also compare results obtained by our model to the results of a real cascade that occurred during a major blackout in the San Diego area on Sept. 2011. The analysis and results presented in this paper will have implications both on the design of new power grids and on identifying the locations for shielding, strengthening, and monitoring efforts in grid upgrades.

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