Showing papers in "IEEE Transactions on Control of Network Systems in 2018"

Harnessing Smoothness to Accelerate Distributed Optimization

Abstract: There has been a growing effort in studying the distributed optimization problem over a network. The objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. The literature has developed consensus-based distributed (sub)gradient descent (DGD) methods and has shown that they have the same convergence rate $O(\frac{\log t}{\sqrt{t}})$ as the centralized (sub)gradient methods (CGD), when the function is convex but possibly nonsmooth. However, when the function is convex and smooth, under the framework of DGD, it is unclear how to harness the smoothness to obtain a faster convergence rate comparable to CGD's convergence rate. In this paper, we propose a distributed algorithm that, despite using the same amount of communication per iteration as DGD, can effectively harnesses the function smoothness and converge to the optimum with a rate of $O(\frac{1}{t})$ . If the objective function is further strongly convex, our algorithm has a linear convergence rate. Both rates match the convergence rate of CGD. The key step in our algorithm is a novel gradient estimation scheme that uses history information to achieve fast and accurate estimation of the average gradient. To motivate the necessity of history information, we also show that it is impossible for a class of distributed algorithms like DGD to achieve a linear convergence rate without using history information even if the objective function is strongly convex and smooth.

Structure Learning in Power Distribution Networks

Abstract: 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.

An Adaptive Switched Control Approach to Heterogeneous Platooning With Intervehicle Communication Losses

Abstract: The advances in distributed intervehicle communication networks have stimulated a fruitful line of research in cooperative adaptive cruise control (CACC). In CACC, individual vehicles, grouped into platoons, must automatically adjust their own speed using on-board sensors and communication with the preceding vehicle so as to maintain a safe intervehicle distance. However, a crucial limitation of the state of the art of this control scheme is that the string stability of the platoon can be proven only when the vehicles in the platoon have identical driveline dynamics and perfect engine performance (homogeneous platoon), and possibly an ideal communication channel. This paper proposes a novel CACC strategy that overcomes the homogeneity assumption and that is able to adapt its action and achieve string stability even for uncertain heterogeneous platoons. Furthermore, in order to handle the inevitable communication losses, we formulate an extended average dwell-time framework and design an adaptive switched control strategy, which activates an augmented CACC or an augmented adaptive cruise control strategy depending on communication reliability. Stability is proven analytically and simulations are conducted to validate the theoretical analysis.

Detection Against Linear Deception Attacks on Multi-Sensor Remote State Estimation

Abstract: In this paper, a security problem in cyberphysical systems (CPS) is studied. A remote state estimation process using multiple sensors is considered. The measurement innovation packets from each sensor, which may be modified by a malicious attacker, are sent to a remote fusion center through wireless communication channels. To avoid being detected by typical bad data detectors at the remote estimator's side, the attacker would maintain the statistical properties of the measurements. Based on the information extracted from the trusted sensors and the correlations between the trusted sensors and the suspicious sensors, we propose three sequential data verification and fusion procedures for different detection information scenarios. The corresponding impacts of possible attacking patterns on the estimation performance under different detectors are analyzed explicitly. Simulations are provided to illustrate the developed results.

DoS Attack Energy Management Against Remote State Estimation

Abstract: This paper considers a remote state estimation problem, where a sensor measures the state of a linear discrete-time process and has computational capability to implement a local Kalman filter based on its own measurements. The sensor sends its local estimates to a remote estimator over a communication channel that is exposed to a Denial-of-Service (DoS) attacker. The DoS attacker, subject to limited energy budget, intentionally jams the communication channel by emitting interference noises with the purpose of deteriorating estimation performance. In order to maximize attack effect, following the existing answer to “when to attack the communication channel”, in this paper we manage to solve the problem of “how much power the attacker should use to jam the channel in each time”. For the static attack energy allocation problem, when the system matrix is normal, we derive a sufficient condition for when the maximum number of jamming operations should be used. The associated jamming power is explicitly provided. For a general system case, we propose an attack power allocation algorithm and show the computational complexity of the proposed algorithm is not worse than $\mathcal{O}(T)$ , where $T$ is the length of the time horizon considered. When the attack can receive the real-time ACK information, we formulate a dynamic attack energy allocation problem, and transform it to a Markov Decision Process to find the optimal solution.

Differentially Private Distributed Convex Optimization via Functional Perturbation

Abstract: We study a class of distributed convex constrained optimization problems where a group of agents aim to minimize the sum of individual objective functions while each desires that any information about its objective function is kept private. We prove the impossibility of achieving differential privacy using strategies based on perturbing the inter-agent messages with noise when the underlying noise-free dynamics are asymptotically stable. This justifies our algorithmic solution based on the perturbation of individual functions with Laplace noise. To this end, we establish a general framework for differentially private handling of functional data. We further design post-processing steps that ensure the perturbed functions regain the smoothness and convexity properties of the original functions while preserving the differentially private guarantees of the functional perturbation step. This methodology allows us to use any distributed coordination algorithm to solve the optimization problem on the noisy functions. Finally, we explicitly bound the magnitude of the expected distance between the perturbed and true optimizers which leads to an upper bound on the privacy-accuracy tradeoff curve. Simulations illustrate our results.

A Jamming-Resilient Algorithm for Self-Triggered Network Coordination

Abstract: The issue of cyber-security has become ever more prevalent in the analysis and design of cyber-physical systems. In this paper, we investigate self-triggered consensus networks in the presence of communication failures caused by denial-of-service (DoS) attacks. A general framework is considered in which the network links can fail independent of each other. By introducing a notion of persistency-of-communication (PoC), we provide an explicit characterization of DoS frequency and duration under which consensus can be preserved by suitably designing time-varying control and communication policies. An explicit characterization of the effects of DoS on the consensus time is also provided. The considered notion of PoC is compared with classic average connectivity conditions that are found in pure continuous-time consensus networks. Finally, examples are given to substantiate the analysis.

Optimal Attack Strategies Subject to Detection Constraints Against Cyber-Physical Systems

Abstract: This paper studies an attacker against a cyber-physical system (CPS) whose goal is to move the state of a CPS to a target state while ensuring that his or her probability of being detected does not exceed a given bound. The attacker's probability of being detected is related to the non-negative bias induced by his or her attack on the CPS's detection statistic. We formulate a linear quadratic cost function that captures the attacker's control goal and establish constraints on the induced bias that reflect the attacker's detection-avoidance objectives. When the attacker is constrained to be detected at the false alarm rate of the detector, we show that the optimal attack strategy reduces to a linear feedback of the attacker's state estimate. In the case that the attacker's bias is upper bounded by a positive constant, we provide two algorithms—an optimal algorithm and a suboptimal, less computationally intensive algorithm—to find suitable attack sequences. Finally, we illustrate our attack strategies in numerical examples based on a remotely controlled helicopter under attack.

Distributed Adaptive Consensus Disturbance Rejection for Multi-Agent Systems on Directed Graphs

Abstract: This paper considers the distributed consensus disturbance rejection problem for general linear multiagent systems with deterministic disturbances under directed communication graphs. Based on the relative state information of the neighboring agents, the consensus protocols, which consist of two observers, including a state observer and a separate disturbance observer, are designed to guarantee that the consensus error goes to zero with complete disturbance rejection. Furthermore, the state observer is designed in a fully distributed fashion with adaptive coupling gain, which has the advantage that the consensus controller design is independent of the Laplacian matrix associated with the communication network. The distributed observer-based consensus disturbance rejection protocols are further extended to containment control. Finally, an example is provided to demonstrate the effectiveness of the proposed strategies.

Event-Triggered Quantized Communication-Based Distributed Convex Optimization

Abstract: A NOVEL distributed algorithm based on multiple agents with continuous-time dynamics is proposed for a convex optimization problem where the objective function is the summation of local objective functions and the state of each agent is subject to a convex constraint set. Considering the limited bandwidth of the communication channels, we introduce a dynamic quantizer for each agent. To further save on communication costs, we develop an event-based broadcasting scheme for each agent. In comparison with algorithms that rely on continuous communication, the proposed algorithm serves to save communication expenditure by exploiting temporal and spatial aspects. Though a joint design of dynamic quantizers and event-trigger functions are under mild conditions, the states of the agents asymptotically approach the global optimal point with an adjustable error bound without incurring Zeno behavior.

Multiagent Decision-Making Dynamics Inspired by Honeybees

Abstract: When choosing between candidate nest sites, a honeybee swarm reliably chooses the most valuable site and even when faced with the choice between near-equal value sites, it makes highly efficient decisions. Value-sensitive decision-making is enabled by a distributed social effort among the honeybees, and it leads to decision-making dynamics of the swarm that are remarkably robust to perturbation and adaptive to change. To explore and generalize these features to other networks, we design distributed multiagent network dynamics that exhibit a pitchfork bifurcation, ubiquitous in biological models of decision-making. Using tools of nonlinear dynamics, we show how the designed agent-based dynamics recover the high performing value-sensitive decision-making of the honeybees and rigorously connect an investigation of mechanisms of animal group decision-making to systematic, bioinspired control of multiagent network systems. We further present a distributed adaptive bifurcation control law and prove how it enhances the network decision-making performance beyond that observed in swarms.

Epidemic Processes Over Time-Varying Networks

Abstract: The spread of viruses in biological networks, computer networks, and human contact networks can have devastating effects; developing and analyzing mathematical models of these systems can provide insights that lead to long-term societal benefits. Prior research has focused mainly on network models with static graph structures; however, the systems being modeled typically have dynamic graph structures. In this paper, we consider virus spread models over networks with dynamic graph structures, and we investigate the behavior of these systems. We perform a stability analysis of epidemic processes over time-varying networks, providing sufficient conditions for convergence to the disease-free equilibrium (the origin, or healthy state), in both the deterministic and stochastic cases. We present simulation results and discuss quarantine control via simulation.

Power Grid State Estimation Following a Joint Cyber and Physical Attack

Abstract: This paper focuses on joint cyber and physical attacks on power grids and presents methods to retrieve the grid state information following such an attack. We consider a model where an adversary attacks a zone by physically disconnecting some of its power lines and blocking the information flow from the zone to the grid's control center. We use tools from linear algebra and graph theory and leverage the properties of the linearized power flow model to develop methods for information recovery. Using information observed outside the attacked zone, these methods recover information about the disconnected lines and the phase angles at the buses . We identify sufficient conditions on the zone structure and constraints on the attack characteristics such that these methods can recover the information. We also show that it is NP-hard to find an approximate solution to the problem of partitioning the power grid into the minimum number of attack-resilient zones. However, since power grids can often be represented by planar graphs, we develop a constant approximation partitioning algorithm for these graphs and numerically demonstrate its performance on real power grids.

Distributed Coordination of Multiple Unknown Euler-Lagrange Systems

Abstract: A robust consensus tracking problem is addressed for multiple unknown Euler-Lagrange systems where only a subset of the agents is informed of the desired time-varying trajectory. Challenging unstructured uncertainties, including unknown nonlinear dynamics and disturbances, are considered in the agent dynamics. A model-free, identifier-based, continuous, distributed robust control method is designed to solve this problem under both undirected and directed graphs. The control inputs and coupling gains depend only on local information and the consensus tracking errors are proven to converge to zero asymptotically. Under an undirected graph, a distributed nonlinear identifier is developed for each agent to compensate for the unknown nonlinear dynamics and disturbances. Based on this identifier, a continuous distributed control law is designed to enable asymptotic robust consensus tracking. By selecting the gains of the designed controller according to the derived conditions, closed-loop stability is proven using graph theory and Lyapunov analysis. Furthermore, the directed graph case is investigated via a distributed two-layer coordination scheme in which a model-free continuous distributed controller is designed by using information obtained from a distributed leader estimator. Numerical simulation results are given to illustrate the effectiveness of the proposed methods.

Secure State Estimation and Control for Cyber Security of the Nonlinear Power Systems

Abstract: We focus on securely estimating the state of a nonlinear dynamical system from a set of corrupted measurements for two classes of nonlinear systems, and propose a technique that enables us to perform secure state estimation for those systems We then illustrate how the proposed nonlinear secure state estimation technique can be used to perform estimation in the cyber layer of interconnected power systems under cyber-physical attacks and communication failures In particular, we focus on an interconnected power system comprised of several synchronous generators, transmission lines, loads, and energy storage units, and propose a secure estimator that allows us to securely estimate the dynamic states of the power network Finally, we numerically demonstrate the effectiveness of the proposed secure estimation algorithm, and show that the algorithm enables the cyber layer to accurately reconstruct the attack signals

Adaptive Reliable Coordination Control for Linear Agent Networks With Intermittent Communication Constraints

Abstract: This paper studies the adaptive reliable coordination control problem for a class of multiagent systems under time-varying topology in the presence of intermittent communication constraints and actuator faults. The underlying communication network is assumed to switch among finite undirected connected graphs. A new fault-tolerant consensus tracking control scheme is presented by utilizing the distributed adaptive technique. In contrast to the existing results on the cooperative tracking control problem against intermittent communications, no assumption on zero input of the leader is made in the discussions. By introducing the estimates to account for the unknown faulty efficiency factors and the bounds of the leader's control input signal, the local controller parameters are updated in individual nodes. Also, a novel topology-allocation-dependent average dwell time approach is proposed to deal with network switching jumps effectively. By combining multiple Lyapunov functions method and algebraic graph theory, it is shown that the global tracking errors are uniformly ultimately bounded even in the cases of intermittent communication constraints and actuator failures. Finally, the simulation study demonstrates the effectiveness of the proposed reliable coordination control strategy.

Resilient First-Order Consensus and Weakly Stable, Higher Order Synchronization of Continuous-Time Networked Multiagent Systems

Abstract: Local interaction rules for consensus and synchronization are vital for many applications in distributed control of cyber-physical systems. However, most research in this area assumes all nodes (or agents) in the networked system cooperate. This paper considers local interaction rules for resilient first-order consensus and weakly stable, higher order synchronization whenever some of the agents in the network are Byzantine-like adversaries defined in a continuous-time setting. The normal agents have identical dynamics modeled by continuous-time, linear, time-invariant, weakly stable systems. Agents in the networked system influence one another by sharing state or output information according to a directed, time-varying graph. We present a resilient consensus protocol as well as dynamic state and output feedback control laws for the normal agents, to achieve the resilient consensus and synchronization objectives, respectively. We characterize the required network topologies using the property of network robustness. We demonstrate the results in simulation examples to illustrate the resilient synchronization output feedback control law.

On Regret-Optimal Learning in Decentralized Multiplayer Multiarmed Bandits

Abstract: We consider the problem of learning in single-player and multiplayer multiarmed bandit models. Bandit problems are classes of online learning problems that capture exploration versus exploitation tradeoffs. In a multiarmed bandit model, players can pick among many arms, and each play of an arm generates an i.i.d. reward from an unknown distribution. The objective is to design a policy that maximizes the expected reward over a time horizon for a single-player setting and the sum of expected rewards for the multiplayer setting. In the multiplayer setting, arms may give different rewards to different players. There is no separate channel for coordination among the players. Any attempt at communication is costly and adds to regret. We propose two decentralizable policies, $\tt E^3$ ( $\tt E$ - $\tt cubed$ ) and $\tt E^3$ - $\tt TS$ , that can be used in both single-player and multiplayer settings. These policies are shown to yield expected regret that grows, at most, as $O(\log ^{1+\delta } T)$ (and $O(\log T)$ under some assumption). It is well known that $O(\log T)$ is the lower bound on the rate of growth of regret even in a centralized case. The proposed algorithms improve on prior work where regret grew at $O(\log ^2 T)$ . More fundamentally, these policies address the question of additional cost incurred in decentralized online learning, suggesting that there is, at most, an $\delta$ -factor cost in terms of the order of regret. This solves a problem of relevance in many domains and had been open for a while.

Energy-Optimal Pump Scheduling and Water Flow

Abstract: This paper focuses on the optimal operation of water supply networks. We model water supply networks using hydraulic constraints, and formulate a joint optimal pump scheduling and water flow problem (OWF) using the hydraulic characteristics of variable speed pumps. OWF is a mixed-integer nonlinear program . This problem is nonconvex, and hence NP-hard. To compute an exact solution of OWF, we first focus on the feasibility region of OWF, and propose a mixed-integer second-order cone relaxation for the feasibility region of OWF. We prove that the proposed relaxation is exact for several relevant network topologies. We then focus on the objective function in OWF, and show that for some energy metrics, OWF can be transformed into a mixed-integer second-order cone program . Furthermore, we propose an ADMM-based algorithm to compute suboptimal solutions to OWF and lower bounds on the optimal value of the objective in OWF when the objective function is nonconvex. Finally, we consider a real-world water network, and demonstrate the effectiveness of the proposed relaxation in computing the optimal pump schedules and water flows.

A Distributed, Asynchronous, and Incremental Algorithm for Nonconvex Optimization: An ADMM Approach

Abstract: The alternating direction method of multipliers (ADMM) has been popular for solving many signal processing problems, convex or nonconvex. In this paper, we study an asynchronous implementation of ADMM for solving a nonconvex nonsmooth optimization problem, whose objective is the sum of a number of component functions. The proposed algorithm allows the problem to be solved in a distributed, asynchronous, and incremental manner. First, the component functions can be distributed to different computing nodes, which perform the updates asynchronously without coordinating with each other. Two sources of asynchrony are covered by our algorithm: One is caused by the heterogeneity of the computational nodes and the other arises from unreliable communication links. Second, the algorithm can be viewed as implementing an incremental algorithm where at each step the (possibly delayed) gradients of only a subset of component functions are updated. We show that when certain bounds are imposed on the level of asynchrony, the proposed algorithm converges to the set of stationary solutions (resp. optimal solutions) for the nonconvex (resp. convex) problem, with a global sublinear rate.

Improving Network Connectivity and Robustness Using Trusted Nodes With Application to Resilient Consensus

Abstract: To observe and control a networked system, especially in failure-prone circumstances, it is imperative that the underlying network structure be robust against node or link failures. A common approach for increasing network robustness is redundancy: deploying additional nodes and establishing new links between nodes, which could be prohibitively expensive. This paper addresses the problem of improving structural robustness of networks without adding extra links. The main idea is to ensure that a small subset of nodes, referred to as the trusted nodes, remains intact and functions correctly at all times. We extend two fundamental metrics of structural robustness with the notion of trusted nodes, network connectivity, and $r$ -robustness, and then show that by controlling the number and location of trusted nodes, any desired connectivity and robustness can be achieved without adding extra links. We study the complexity of finding trusted nodes and construction of robust networks with trusted nodes. Finally, we present a resilient consensus algorithm with trusted nodes and show that, unlike existing algorithms, resilient consensus is possible in sparse networks containing few trusted nodes.

Adaptive Consensus in Leader-Following Networks of Heterogeneous Linear Systems

Abstract: This paper investigates a general model of heterogeneous multiagent systems with different individual adaptation structures and input constraints, and proposes an effective distributed adaptation protocol for compensating the effects of differences in system matrices and solving the leader-following consensus problem in such a model. It is generally assumed that state outputs are the only information transmitted over networks, and relative states between neighboring agents are locally available to the linked agents. Sufficient conditions are established for adaptive state consensus in terms of rooted interaction topologies, and a simulation of synchronizing harmonic oscillators is given to demonstrate the effectiveness of the proposed results.

Distributed Optimal Load Frequency Control with Non-Passive Dynamics

Abstract: Motivated by an increase of renewable energy sources, we propose a distributed optimal load frequency control scheme achieving frequency regulation and economic dispatch. Based on an energy function of the power network, we derive an incremental passivity property for a well-known nonlinear structure preserving network model, differentiating between generator and load buses. Exploiting this property, we design distributed controllers that adjust the power generation. Notably, we explicitly include the turbine-governor dynamics, where first-order and the widely used second-order dynamics are analyzed in a unifying way. Due to the non-passive nature of the second-order turbine-governor dynamics, incorporating them is challenging, and we develop a suitable dissipation inequality for the interconnected generator and turbine-governor. This allows us to include the generator side more realistically in the stability analysis of optimal load frequency control than was previously possible.

Compositional Abstraction for Networks of Control Systems: A Dissipativity Approach

Abstract: In this paper, we propose a compositional scheme for the construction of abstractions for networks of control systems by using the interconnection matrix and joint dissipativity-type properties of subsystems and their abstractions. In the proposed framework, the abstraction, itself a control system (possibly with a lower dimension), can be used as a substitution of the original system in the controller design process. Moreover, we provide a procedure for constructing abstractions of a class of nonlinear control systems by using the bounds on the slope of system nonlinearities. We illustrate the proposed results on a network of linear control systems by constructing its abstraction in a compositional way without requiring any condition on the number or gains of the subsystems. We use the abstraction as a substitute to synthesize a controller enforcing a certain linear temporal logic specification. This example particularly elucidates the effectiveness of dissipativity-type compositional reasoning for large-scale systems.

Fault-Tolerant Clock Synchronization Over Unreliable Channels in Wireless Sensor Networks

Abstract: In this paper, we study a distributed approach based on consensus algorithms for clock synchronization in wireless sensor networks. The sensor nodes face two types of uncertainties. One is that some of the nodes in the network can be faulty and transmit arbitrary signals by not following the given protocol; similar effects may be caused by false data injection by an external malicious attacker. The other is that the communication is unreliable and the packets exchanged may become lost. To deal with these uncertainties, we propose a resilient consensus-type algorithm based on the so-called mean subsequence reduced (MSR) technique, where each normal node ignores the outliers in the clock data collected from its neighbors and makes updates using data from the past if new data have not arrived yet. We establish network connectivity conditions in terms of graph robustness for the MSR algorithm to attain resilient properties.

Output Synchronization in Coupled Neural Networks With and Without External Disturbances

Abstract: This paper studies the output synchronization of coupled neural networks (CNNs) as well as the effects of external disturbances. By employing matrix theory and Barbalat's Lemma, several output synchronization criteria are presented for CNNs with directed and undirected topologies, respectively. Moreover, in order to ensure the output synchronization of CNNs, two adaptive schemes to adjust the coupling weights are designed. On the other hand, we, respectively, analyze the $\mathcal {H}_{\infty }$ output synchronization of directed and undirected CNNs with external disturbances, and two adaptive strategies for updating the coupling weights are designed to guarantee the $\mathcal {H}_{\infty }$ output synchronization of CNNs. Finally, two examples of CNNs are also given to verify the proposed output synchronization criteria.

Cloud-Enabled Differentially Private Multiagent Optimization With Constraints

Abstract: We present an optimization framework for solving multiagent convex programs subject to inequality constraints while keeping the agents’ state trajectories private. Each agent has an objective function depending only upon its own state and the agents are collectively subject to global constraints. The agents do not directly communicate with each other but instead route messages through a trusted cloud computer. The cloud adds noise to data being sent to the agents in accordance with the framework of differential privacy and, thus, keeps each agent's state trajectory private from all other agents and any eavesdroppers. This private problem can be viewed as a stochastic variational inequality, and it is solved using a projection-based method for solving variational inequalities that resemble a noisy primal-dual gradient algorithm. Convergence of the optimization algorithm in the presence of noise is proven, and a quantifiable tradeoff between privacy and convergence is extracted from this proof. Simulation results are provided that demonstrate numerical convergence for both $\epsilon$ -differential privacy and $(\epsilon, \delta)$ -differential privacy.

Conic Relaxations for Power System State Estimation With Line Measurements

Abstract: This paper deals with the nonconvex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows (PFs), for which the existing methods lack guaranteed convergence and theoretical analysis. Motivated by these limitations, we propose a novel convexification framework for the PSSE problem using semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations. We first study a related PF problem as the noiseless counterpart, which is cast as a constrained minimization program by adding a suitably designed objective function. We study the performance of the proposed framework in the case where the set of measurements includes: 1) nodal voltage magnitudes and 2) branch active PFs over at least a spanning tree of the network. It is shown that the SDP and SOCP relaxations both recover the true PF solution as long as the voltage angle difference across each line of the network is not too large (e.g., less than $90^{\circ }$ for lossless networks). By capitalizing on this result, penalized SDP and SOCP problems are designed to solve the PSSE, where a penalty based on the weighted least absolute value is incorporated for fitting noisy measurements with possible bad data. Strong theoretical results are derived to quantify the optimal solution of the penalized SDP problem, which is shown to possess a dominant rank-one component formed by lifting the true voltage vector. An upper bound on the estimation error is also derived as a function of the noise power, which decreases exponentially fast as the number of measurements increases. Numerical results on benchmark systems, including a 9241-bus European system, are reported to corroborate the merits of the proposed convexification framework.

Robustness of Leader–Follower Networked Dynamical Systems

Abstract: We present a graph–theoretic approach to analyze the robustness of leader–follower consensus dynamics to disturbances and time delays. Robustness to disturbances is captured via the system $\mathcal{H}_2$ and $\mathcal{H}_{\infty }$ norms, and robustness to time delay is defined as the maximum-allowable delay for the system to remain asymptotically stable. Our analysis is built on understanding certain spectral properties of the grounded Laplacian matrix that play a key role in such dynamics. Specifically, we give graph–theoretic bounds on the extreme eigenvalues of the grounded Laplacian matrix that quantify the impact of disturbances and time delays on the leader–follower dynamics. We then provide tight characterizations of these robustness metrics in Erdős–R $\acute{e}$ nyi random graphs and random d-regular graphs. Finally, we view robustness to disturbances and time delay as network centrality metrics, and provide conditions under which a leader in a network optimizes each robustness objective. Furthermore, we propose a sufficient condition under which a single leader optimizes both robustness objectives simultaneously.

Scheduling of EV Battery Swapping—Part I: Centralized Solution

Abstract: We formulate an optimal scheduling problem for battery swapping that assigns to each electric vehicle (EV) a best battery station to swap its depleted battery based on its current location and state of charge. The schedule aims to minimize a weighted sum of EVs’ travel distance and electricity generation cost over both station assignments and power flow variables, subject to EV range constraints, grid operational constraints, and ac power flow equations. To deal with the nonconvexity of power flow equations and the binary nature of station assignments, we propose a solution based on second-order cone programming (SOCP) relaxation of optimal power flow and generalized Benders decomposition. When the SOCP relaxation is exact, this approach computes a global optimum. We evaluate the performance of the proposed algorithm through simulations. The algorithm requires global information and is suitable for cases where the distribution grid, battery stations, and EVs are managed centrally by the same operator. In Part II of this paper, we develop distributed solutions for cases where they are operated by different organizations that do not share private information.