This paper studies the problem of frequency regulation in power grids under unknown and possible time-varying load changes, while minimizing the generation costs. We formulate this problem as an output agreement problem for distribution networks and address it using incremental passivity and distributed internal-model-based controllers. Incremental passivity enables a systematic approach to study convergence to the steady state with zero frequency deviation and to design the controller in the presence of time-varying voltages, whereas the internal-model principle is applied to tackle the uncertain nature of the loads.

This paper presents a class of passivity-based cooperative control problems that have an explicit connection to convex network optimization problems. The new notion of maximal equilibrium independent passivity is introduced and it is shown that networks of systems possessing this property asymptotically approach the solutions of a dual pair of network optimization problems, namely an optimal potential and an optimal flow problem. This connection leads to an interpretation of the dynamic variables, such as system inputs and outputs, to variables in a network optimization framework, such as divergences and potentials, and reveals that several duality relations known in convex network optimization theory translate directly to passivity-based cooperative control problems. The presented results establish a strong and explicit connection between passivity-based cooperative control theory on the one side and network optimization theory on the other, and they provide a unifying framework for network analysis and optimal design. The results are illustrated on a nonlinear traffic dynamics model that is shown to be asymptotically clustering. IntroductionOne of the most profound concepts in mathematics is the notion of duality. In many ways, duality theory is the mathematical answer to the idiom "there are two sides to every coin." This powerful concept manifests itself across many mathematical disciplines, but perhaps the most elegant and complete notion of duality is the celebrated Lagrange duality in convex optimization theory [1]. One of the most complete expositions of this duality theory relates to a class of optimization problems over networks, generally known as network optimization [2], [3]. In [2], a unifying framework for network optimization was established, with the key elements being a pair of dual optimization problems: the optimal network flow problem and the optimal potential problem. This dual pair of optimization problems characterizes the majority of network decision problems. The notion of duality also has a long history within the theory of control systems, as control problems are often intimately related to optimization problems and their respective duals have again a controls interpretation, see e.g. [4].A recent trend in modern control theory is the study of cooperative control problems amongst groups of dynamical systems that interact over an information exchange network. A fundamental goal for the analysis of these systems is to reveal the interplay between properties of the individual dynamic agents, the underlying network topology, and the interaction protocols that influence the functionality of the overall system [5]. Amongst the numerous control theoretic approaches being pursued to define a general theory for networks of dynamical systems, passivity [6] takes an outstanding role; see e.g., [7]. The conceptual idea underlying passivity-based cooperative control is to separate the network analysis and synthesis into two layers. On the systems layer, each dynamical system comprising the network ...

a b s t r a c tThis paper studies the problem of output agreement in networks of nonlinear dynamical systems under time-varying disturbances, using dynamic diffusive couplings. Necessary conditions are derived for general networks of nonlinear systems, and these conditions are explicitly interpreted as conditions relating the node dynamics and the network topology. For the class of incrementally passive systems, necessary and sufficient conditions for output agreement are derived. The approach proposed in the paper lends itself to solve flow control problems in distribution networks. As a first case study, the internal model approach is used for designing a controller that achieves an optimal routing and inventory balancing in a dynamic transportation network with storage and time-varying supply and demand. It is in particular shown that the time-varying optimal routing problem can be solved by applying an internal model controller to the dual variables of a certain convex network optimization problem. As a second case study, we show that droop-controllers in microgrids have also an interpretation as internal model controllers.

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This work considers the robustness of uncertain consensus networks. The first set of results studies the stability properties of consensus networks with negative edge weights. We show that if either the negative weight edges form a cut in the graph, or any single negative edge weight has magnitude less than the inverse of the effective resistance between the two incident nodes, then the resulting network is unstable. These results are then applied to analyze the robustness properties of the consensus network with additive but bounded perturbations of the edge weights. It is shown that the small-gain condition is related again to cuts in the graph and effective resistance. For the single edge case, the small-gain condition is also shown to be exact. The results are then extended to consensus networks with non-linear couplings.