This paper considers distributed online optimization with time-varying coupled inequality constraints. The global objective function is composed of local convex cost and regularization functions and the coupled constraint function is the sum of local convex functions. Adistributed online primal-dual dynamic mirror descent algorithm is proposed to solve this problem, where the local cost, regularization, and constraint functions are held privately and revealed only after each time slot. Without assuming Slater's condition, we first derive regret and constraint violation bounds for the algorithm and show how they depend on the stepsize sequences, the accumulated dynamic variation of the comparator sequence, the number of agents, and the network connectivity. As a result, under some natural decreasing stepsize sequences, we prove that the algorithm achieves sublinear dynamic regret and constraint violation if the accumulated dynamic variation of the optimal sequence also grows sublinearly. We also prove that the algorithm achieves sublinear static regret and constraint violation under mild conditions. Assuming Slater's condition, we show that the algorithm achieves smaller bounds on the constraint violation. In addition, smaller bounds on the static regret are achieved when the objective function is strongly convex. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

In distributed optimization of multi-agent systems, agents cooperate to minimize a global function which is a sum of local objective functions. Motivated by applications including power systems, sensor networks, smart buildings, and smart manufacturing, various distributed optimization algorithms have been developed. In these algorithms, each agent performs local computation based on its own information and information received from its neighboring agents through the underlying communication network, so that the optimization problem can be solved in a distributed manner. This survey paper aims to offer a detailed overview of existing distributed optimization algorithms and their applications in power systems. More specifically, we first review discrete-time and continuous-time distributed optimization algorithms for undirected graphs. We then discuss how to extend these algorithms in various directions to handle more realistic scenarios. Finally, we focus on the application of distributed optimization in the optimal coordination of distributed energy resources. 

We propose two novel dynamic event-triggered control laws to solve the average consensus problem for first-order continuous-time multiagent systems over undirected graphs. Compared with the most existing triggering laws, the proposed laws involve internal dynamic variables, which play an essential role in guaranteeing that the triggering time sequence does not exhibit Zeno behavior. Moreover, some existing triggering laws are special cases of ours. For the proposed self-triggered algorithm, continuous agent listening is avoided as each agent predicts its next triggering time and broadcasts it to its neighbors at the current triggering time. Thus, each agent only needs to sense and broadcast at its triggering times, and to listen to and receive incoming information from its neighbors at their triggering times. It is proved that the proposed triggering laws make the state of each agent converge exponentially to the average of the agents' initial states if and only if the underlying graph is connected. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Nonlinearities are present in all real applications. Two types of general nonlinear consensus protocols are considered in this paper, namely, the systems with nonlinear communication and actuator constraints. The solutions of the systems are understood in the sense of Filippov to handle the possible discontinuity of the controllers. For each case, we prove the asymptotic stability of the systems with minimal assumptions on the nonlinearity, for both directed and undirected graphs. These results extend the literature to more general nonlinear dynamics and topologies. As applications of established theorems, we interpret the results on quantized consensus protocols.

In this paper, we propose a fully distributed algorithm for second-order continuous-time multi-agent systems to solve the distributed optimization problem. The global objective function is a sum of private cost functions associated with the individual agents and the interaction between agents is described by a weighted undirected graph. We show the exponential convergence of the proposed algorithm if the underlying graph is connected, each private cost function is locally gradient-Lipschitz- continuous, and the global objective function is restricted strongly convex with respect to the global minimizer. Moreover, to reduce the overall need of communication, we then propose a dynamic event-triggered communication mechanism that is free of Zeno behavior. It is shown that the exponential convergence is achieved if the private cost functions are also globally gradient-Lipschitz- continuous. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.

In this paper, we consider the distributed optimization problem, whose objective is to minimize the global objective function, which is the sum of local convex objective functions, by using local information exchange. To avoid continuous communication among the agents, we propose a distributed algorithm with a dynamic event-triggered communication mechanism. We show that the distributed algorithm with the dynamic event-triggered communication scheme converges to the global minimizer exponentially, if the underlying communication graph is undirected and connected. Moreover, we show that the event-triggered algorithm is free of Zeno behavior. For a particular case, we also explicitly characterize the lower bound for inter-event times. The theoretical results are illustrated by numerical simulations.

This paper presents the formulation and analysis of a fully distributed dynamic event-triggered communication based robust dynamic average consensus algorithm. Dynamic average consensus problem involves a networked set of agents estimating the time-varying average of dynamic reference signals locally available to individual agents. We propose an asymptotically stable solution to the dynamic average consensus problem that is robust to network disruptions. Since this robust algorithm requires continuous communication among agents, we introduce a novel dynamic event-triggered communication scheme to reduce the overall inter-agent communications. It is shown that the event-triggered algorithm is asymptotically stable and free of Zeno behavior. Numerical simulations are provided to illustrate the effectiveness of the proposed algorithm.

This paper studies the synchronization of a finite number of Kuramoto oscillators in a frequency-dependent bidirectional tree network. We assume that the coupling strength of each link in each direction is equal to the product of a common coefficient and the exogenous frequency of its corresponding source oscillator. We derive a sufficient condition for the common coupling strength in order to guarantee frequency synchronization in tree networks. Moreover, we discuss the dependency of the obtained bound on both the graph structure and the way that exogenous frequencies are distributed. Further, we present an application of the obtained result by means of an event-triggered algorithm for achieving frequency synchronization in a star network assuming that the common coupling coefficient is given.

In this paper, we study the consensus problem of linear multi-agent systems. Centralized event-triggered rules are provided so as to reduce the frequency of system's updating. The diffusion coupling feedbacks of each agent are based on the latest observations from its in-neighbours and the system's next observation time is triggered by a criterion based on all agents' information. The scenario of continuous monitoring is first considered, namely all agents' instantaneous states can be observed. It is proved that if the network topology is connected, then this centralized event-triggered coupling strategy can realize consensus exponentially for the linear multi-agent systems. Then the results are extended to discontinuous monitoring, where the system computes its next triggering time in advance without having to observe all agents' states continuously. As a special case, we apply above results to double-integrator consensus problem. One example with numerical simulation are provided to show the effectiveness of the theoretical results.