Mathias Bürger

An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages

The power grid can be regarded as a large interconnected network of different subsystems, called control area’s. In order to guarantee reliable operation the frequency is tightly regulated around its nominal value, e.g. 50 Hz. Automatic regulation of the frequency in power grid is traditionally achieved by primary proportional control (droop-control) and a secondary PI-control on the generators, where economic considerations are largely neglected. Recently we proposed a novel distributed controller in [1] that controls power production such that the frequency is regulated in an economic efficient way. Main advantage of our approach, based on passivity properties of the system, is that it has the potential to deal with more complex models of dynamical networks and fairly rich classes of external perturbations.

A distributed simplex algorithm for degenerate linear programs and multi-agent assignments

In this paper we propose a novel distributed algorithm to solve degenerate linear programs on asynchronous peer-to-peer networks with distributed information structures. We propose a distributed version of the well-known simplex algorithm for general degenerate linear programs. A network of agents, running our algorithm, will agree on a common optimal solution, even if the optimal solution is not unique, or will determine infeasibility or unboundedness of the problem. We establish how the multi-agent assignment problem can be efficiently solved by means of our distributed simplex algorithm. We provide simulations supporting the conjecture that the completion time scales linearly with the diameter of the communication graph.

Dynamic coupling design for nonlinear output agreement and time-varying flow control

This 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.

Duality and network theory in passivity-based cooperative control ☆

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.

A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization

In this paper, we consider a general problem setup for a wide class of convex and robust distributed optimization problems in peer-to-peer networks. In this setup, convex constraint sets are distributed to the network processors who have to compute the optimizer of a linear cost function subject to the constraints. We propose a novel fully distributed and asynchronous algorithm, named cutting-plane consensus, to solve the problem, based on a polyhedral outer approximation of the constraint sets. Processors running the algorithm compute and exchange linear approximations of their locally feasible sets. Independently of the number of processors in the network, each processor stores only a small number of linear constraints, making the algorithm scalable to large networks. The cutting-plane consensus algorithm is presented and analyzed for the general framework. Specifically, we prove the correctness of the algorithm, and show its robustness against communication or processor failures. Then, the cutting-plane consensus algorithm is specified to three different classes of distributed optimization problems, namely 1) inequality constrained problems, 2) robust optimization problems, and 3) almost separable optimization problems. For each one of these problem classes we solve a concrete problem and present computational results. That is, we show how to solve: position estimation in wireless sensor networks, a distributed robust linear program, and a distributed microgrid control problem.

On the Robustness of Uncertain Consensus Networks

This work considers the robustness of uncertain consensus networks. The stability properties of consensus networks with negative edge weights are also examined. We show that the network is unstable if either the negative weight edges form a cut in the graph or any single negative edge weight has a magnitude less than the inverse of the effective resistance between the two incident nodes. These results are then used to analyze the robustness 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 nonlinear couplings.

Robust Constraint Satisfaction for Continuous-Time Nonlinear Systems in Strict Feedback Form

This work is concerned with the design of feedback control laws that guarantee satisfaction of hard output constraints. A recursive procedure is used to derive sufficient conditions for robust constraint satisfaction in input affine nonlinear systems with bounded disturbances or parametric uncertainties. The control objective is achieved by a suitable switching between two control modes, one responsible for stabilization and one for constraint satisfaction.

Internal models for nonlinear output agreement and optimal flow control

Abstract This paper studies the problem of output agreement in networks of nonlinear dynamical systems under time-varying disturbances. Necessary and sufficient conditions for output agreement are derived for the class of incrementally passive systems. Following this, it is shown that the optimal distribution problem in dynamic inventory systems with time-varying supply and demand can be cast as a special version of the output agreement problem. We show in particular that the time-varying optimal distribution problem can be solved by applying an internal model controller to the dual variables of a certain convex network optimization problem.

Hierarchical Clustering of Dynamical Networks Using a Saddle-Point Analysis

This paper studies cluster synchronization in dynamical networks. A class of cooperative dynamical networks that exhibit clustering in their asymptotic behavior is analyzed. The network nodes are equipped with heterogeneous dynamics and interact with a nonlinear and saturated interaction rule. It is proven that cluster synchronization appears asymptotically independent of the initial conditions. The clustering behavior of the dynamic network is shown to correspond to the solution of a static saddle-point problem, enabling a precise characterization of the clustering structure. We show how the clustering structure depends on the relation between the underlying graph, the node dynamics, and the saturation level of the interactions. This interpretation leads to deeper combinatorial insights related to clustering, including a generalization of classical network partitioning problems such as the inhibiting bisection problem, the min s-t-cut problem, and hierarchical clustering analysis. The theoretical results are applied for the analysis of a test-case network, inspired by the IEEE 30-bus system.

Practical synchronization with diffusive couplings

We investigate the problem of synchronizing nonidentical or perturbed nonlinear systems. In the considered setup, the systems are incapable to synchronize under diffusive couplings. Instead, assuming the quad property for each system, we derive conditions under which the synchronization error can be kept arbitrarily small by a proper choice of the interconnection structure. This motivates the definition of practical synchronization as an alternative synchronization notion for nonidentical or perturbed dynamical systems. The presented results are intimately related to synchronization of passive systems, but it is shown that the stronger quad assumption is essential in our framework. The proposed concept of practical synchronization translates directly into a notion of robust synchronization. Beyond that, the results open the way for an investigation of synchronization phenomena on unbalanced graphs, leading to the concept of cluster synchronization.