Real-Time Feedback-Based Optimization of Distribution Grids: A Unified Approach

This paper develops an algorithmic framework for real-time optimization of distribution-level distributed energy resources (DERs). The proposed framework optimizes the operation of both DERs that are individually controllable and groups of DERs (i.e., aggregations) at an electrical point of connection that are jointly controlled. From an electrical standpoint, wye and delta single- and multi-phase connections are accounted for. The algorithm enables (groups of) DERs to pursue given performance objectives, while adjusting their (aggregate) powers to respond to services requested by grid operators and to maintain electrical quantities within engineering limits. The design of the algorithm leverages a time-varying bi-level problem formulation capturing various performance objectives and engineering constraints, and an online implementation of primal-dual projected-gradient methods. The gradient steps are suitably modified to accommodate appropriate measurements from the distribution network and the DERs. By virtue of this approach, the resultant algorithm can cope with inaccuracies in the distribution-system modeling, it avoids pervasive metering to gather the state of non-controllable resources, and it naturally lends itself to a distributed implementation. Analytical stability and convergence claims are established in terms of tracking of the solution of the formulated time-varying optimization problem. The proposed method is tested in a realistic distribution system with real data

Statistical Routing for Multihop Wireless Cognitive Networks

To account for the randomness of propagation channels and interference levels in hierarchical spectrum sharing, a novel approach to multihop routing is introduced for cognitive random access networks, whereby packets are randomly routed according to outage probabilities. Leveraging channel and interference level statistics, the resultant cross-layer optimization framework provides optimal routes, transmission probabilities, and transmit-powers, thus enabling cognizant adaptation of routing, medium access, and physical layer parameters to the propagation environment. The associated optimization problem is non-convex, and hence hard to solve in general. Nevertheless, a successive convex approximation approach is adopted to efficiently find a Karush-Kuhn-Tucker solution. Augmented Lagrangian and primal decomposition methods are employed to develop a distributed algorithm, which also lends itself to online implementation. Enticingly, the fresh look advocated here permeates benefits also to conventional multihop wireless networks in the presence of channel uncertainty.Comment: Accepted for publication on the IEEE Journal on Selected Areas in Communications - Cognitive Radio Series (Nov 2012 Issue

Risk-Constrained Microgrid Reconfiguration Using Group Sparsity

The system reconfiguration task is considered for existing power distribution systems and microgrids, in the presence of renewable-based generation and load foresting errors. The system topology is obtained by solving a chance-constrained optimization problem, where loss-of-load (LOL) constraints and Ampacity limits of the distribution lines are enforced. Similar to various distribution system reconfiguration renditions, solving the resultant problem is computationally prohibitive due to the presence of binary line selection variables. Further, lack of closed form expressions for the joint probability distribution of forecasting errors hinders tractability of LOL constraints. Nevertheless, a convex problem re-formulation is developed here by resorting to a scenario approximation technique, and by leveraging the underlying group-sparsity attribute of currents flowing on distribution lines equipped with tie and sectionalizing switches. The novel convex LOL-constrained reconfiguration scheme can also afford a distributed solution using the alternating direction method of multipliers, to address the case where multi-facilities are managed autonomously from the rest of the system.Comment: The paper will appear in IEEE Transactions on Sustainable Energy (accepted May 2014

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

Online Stochastic Optimization of Networked Distributed Energy Resources

This paper investigates distributed control and incentive mechanisms to coordinate distributed energy resources (DERs) with both continuous and discrete decision variables as well as device dynamics in distribution grids. We formulate a multi-period social welfare maximization problem, and based on its convex relaxation propose a distributed stochastic dual gradient algorithm for managing DERs. We further extend it to an online realtime setting with time-varying operating conditions, asynchronous updates by devices, and feedback being leveraged to account for nonlinear power flows as well as reduce communication overhead. The resulting algorithm provides a general online stochastic optimization algorithm for coordinating networked DERs with discrete power setpoints and dynamics to meet operational and economic objectives and constraints. We characterize the convergence of the algorithm analytically and evaluate its performance numerically

Online Proximal-ADMM For Time-varying Constrained Optimization

This paper considers a convex optimization problem with cost and constraints that evolve over time. The function to be minimized is strongly convex and possibly non-differentiable, and variables are coupled through linear constraints. In this setting, the paper proposes an online algorithm based on the alternating direction method of multipliers(ADMM), to track the optimal solution trajectory of the time-varying problem; in particular, the proposed algorithm consists of a primal proximal gradient descent step and an appropriately perturbed dual ascent step. The paper derives tracking results, asymptotic bounds, and linear convergence results. The proposed algorithm is then specialized to a multi-area power grid optimization problem, and our numerical results verify the desired properties

Regulation of Dynamical Systems to Optimal Solutions of Semidefinite Programs: Algorithms and Applications to AC Optimal Power Flow

This paper considers a collection of networked nonlinear dynamical systems, and addresses the synthesis of feedback controllers that seek optimal operating points corresponding to the solution of network-wide constrained optimization problems. Particular emphasis is placed on the solution of semidefinite programs (SDPs). The design of the feedback controller is grounded on a dual epsilon-subgradient approach, with the dual iterates utilized to dynamically update the dynamical-system reference signals. Global convergence is guaranteed for diminishing stepsize rules, even when the reference inputs are updated at a faster rate than the dynamical-system settling time. The application of the proposed framework to the control of power-electronic inverters in AC distribution systems is discussed. The objective is to bridge the time-scale separation between real-time inverter control and network-wide optimization. Optimization objectives assume the form of SDP relaxations of prototypical AC optimal power flow problems.Comment: This is a longer version of a paper submitted to the 2015 American Control Conference. This version contains proofs and additional numerical result

Dynamic Network Delay Cartography

Path delays in IP networks are important metrics, required by network operators for assessment, planning, and fault diagnosis. Monitoring delays of all source-destination pairs in a large network is however challenging and wasteful of resources. The present paper advocates a spatio-temporal Kalman filtering approach to construct network-wide delay maps using measurements on only a few paths. The proposed network cartography framework allows efficient tracking and prediction of delays by relying on both topological as well as historical data. Optimal paths for delay measurement are selected in an online fashion by leveraging the notion of submodularity. The resulting predictor is optimal in the class of linear predictors, and outperforms competing alternatives on real-world datasets.Comment: Part of this paper has been published in the \emph{IEEE Statistical Signal Processing Workshop}, Ann Arbor, MI, Aug. 201

Optimal Dispatch of Photovoltaic Inverters in Residential Distribution Systems

Low-voltage distribution feeders were designed to sustain unidirectional power flows to residential neighborhoods. The increased penetration of roof-top photovoltaic (PV) systems has highlighted pressing needs to address power quality and reliability concerns, especially when PV generation exceeds the household demand. A systematic method for determining the active- and reactive-power set points for PV inverters in residential systems is proposed in this paper, with the objective of optimizing the operation of the distribution feeder and ensuring voltage regulation. Binary PV-inverter selection variables and nonlinear power-flow relations render the novel optimal inverter dispatch problem nonconvex and NP-hard. Nevertheless, sparsity-promoting regularization approaches and semidefinite relaxation techniques are leveraged to obtain a computationally feasible convex reformulation. The merits of the proposed approach are demonstrated using real-world PV-generation and load-profile data for an illustrative low-voltage residential distribution system

Photovoltaic Inverter Controllers Seeking AC Optimal Power Flow Solutions

This paper considers future distribution networks featuring inverter-interfaced photovoltaic (PV) systems, and addresses the synthesis of feedback controllers that seek real- and reactive-power inverter setpoints corresponding to AC optimal power flow (OPF) solutions. The objective is to bridge the temporal gap between long-term system optimization and real-time inverter control, and enable seamless PV-owner participation without compromising system efficiency and stability. The design of the controllers is grounded on a dual epsilon-subgradient method, and semidefinite programming relaxations are advocated to bypass the non-convexity of AC OPF formulations. Global convergence of inverter output powers is analytically established for diminishing stepsize rules and strictly convex OPF costs for cases where: i) computational limits dictate asynchronous updates of the controller signals, and ii) inverter reference inputs may be updated at a faster rate than the power-output settling time. Although the focus is on PV systems, the framework naturally accommodates different types of inverter-interfaced energy resources.Comment: Accepted for publication on IEEE Transactions on Power System