On Factorization, Interconnection and Reduction of Collocated Port-Hamiltonian Systems

Based on a geometric interpretation of nonlinear balanced reduction some implications of this approach are analyzed in the case of collocated port-Hamiltonian systems which have a certain balance in its structure. Furthermore, additional examples of reduction for this class of systems are presented.

Low computational complexity model reduction of power systems with preservation of physical characteristics

A data-driven algorithm recently proposed to solve the problem of model reduction by moment matching is extended to multi-input, multi-output systems. The algorithm is exploited for the model reduction of large-scale interconnected power systems and it offers, simultaneously, a low computational complexity approximation of the moments and the possibility to easily enforce constraints on the reduced order model. This advantage is used to preserve selected slow and poorly damped modes. The preservation of these modes has been shown to be important from a physical point of view and in obtaining an overall good approximation. The problem of the choice of the socalled tangential directions is also analyzed. The algorithm and the resulting reduced order model are validated with the study of the dynamic response of the NETS-NYPS benchmark system (68-Bus, 16-Machine, 5-Area) to multiple fault scenarios

Model Reduction Methods for Complex Network Systems

Network systems consist of subsystems and their interconnections, and provide a powerful framework for analysis, modeling and control of complex systems. However, subsystems may have high-dimensional dynamics, and the amount and nature of interconnections may also be of high complexity. Therefore, it is relevant to study reduction methods for network systems. An overview on reduction methods for both the topological (interconnection) structure of the network and the dynamics of the nodes, while preserving structural properties of the network, and taking a control systems perspective, is provided. First topological complexity reduction methods based on graph clustering and aggregation are reviewed, producing a reduced-order network model. Second, reduction of the nodal dynamics is considered by using extensions of classical methods, while preserving the stability and synchronization properties. Finally, a structure-preserving generalized balancing method for simplifying simultaneously the topological structure and the order of the nodal dynamics is treated.Comment: To be published in Annual Review of Control, Robotics, and Autonomous System

Model order reduction for coupled systems using low-rank approximations

This thesis presents the result of the study on model order reduction (MOR) methods, that can be applied to coupled systems. The goal of the research was to develop reduction techniques, that preserve special properties of coupled or interconnected system, e.g. block-structure of the underlying matrices. On the other hand, the new techniques should also be able to benefit from the knowledge, that the system they are applied to, consists of two (or more) sub-systems or describes some phenomena in different physical domains. As a result of this study, two main approaches are proposed. Their general description is given in the following paragraphs. First, the Separate Bases Reduction (SBR) algorithm is developed, which is a projection based MOR technique that uses Krylov subspaces as reduction bases. The novelty of this method is that SBR algorithm, unlike standard reduction methods designed for coupled problems, uses an uncoupled formulation of the system. In other words, an appropriate Krylov subspace is built for each of the sub-system constituting the interconnected system. As a result, the computational costs of application of the SBR algorithm, with respect to time and memory storage needed for calculations, is lower than in case of MOR methods that use the coupled formulation of the system. Moreover, the blockdiagonal form of the reduction matrices allows for preservation of the block-structure of the system matrices and keeps the sub-systems (or different physical domains) still recognizable in the reduced-order model. The SBR algorithm was successfully applied to a few test cases, resulting in the reduced systems that approximate the original ones with accuracy comparable to the accuracy of systems reduced by means of other blockstructure preserving MOR methods. The second topic of the research focuses on the couplings between the sub-systems. Here, the off-diagonal blocks of the system matrices that correspond to the couplings, are approximated by matrices of lower rank. As a main tool, generalized singular value decomposition (GSVD) is used, which allows to find the most important components of a coupling block with respect to one of the sub-systems. Although this method does not reduce the dimension of the considered problem, it gives benefits if used before application of a MOR technique. First of all, the use of low-rank approximations of the coupling blocks can decrease the computational costs of the the Krylov subspaces construction needed for reduction. If the couplings can be approximated by sufficiently low-rank blocks, the necessary matrix inverse calculation can be performed cheaper, by application of the Sherman-Morrison formula. Moreover, the undesired growth of the reduction bases, in case of use of the SBR algorithm to sub-systems with many inputs (outputs), can be lowered by use of only dominant components of the input (output) space. The conducted experiments showed, that for some cases, the number of the components used to define the couplings can be significantly reduced

Reduced-order modeling of large-scale network systems

Large-scale network systems describe a wide class of complex dynamical systems composed of many interacting subsystems. A large number of subsystems and their high-dimensional dynamics often result in highly complex topology and dynamics, which pose challenges to network management and operation. This chapter provides an overview of reduced-order modeling techniques that are developed recently for simplifying complex dynamical networks. In the first part, clustering-based approaches are reviewed, which aim to reduce the network scale, i.e., find a simplified network with a fewer number of nodes. The second part presents structure-preserving methods based on generalized balanced truncation, which can reduce the dynamics of each subsystem.Comment: Chapter 11 in the book Model Order Reduction: Volume 3 Application

Model reduction of synchronized homogeneous Lur'e networks with incrementally sector-bounded nonlinearities

This paper proposes a model order reduction scheme that reduces the complexity of diffusively coupled homogeneous Lur'e systems. We aim to reduce the dimension of each subsystem and meanwhile preserve the synchronization property of the overall network. Using the Laplacian spectral radius, we characterize the robust synchronization of the Lur'e network by a linear matrix inequality (LMI), whose solutions then are treated as generalized Gramians for the balanced truncation of the linear component of each Lur'e subsystem. It is verified that, with the same communication topology, the resulting reduced-order network system is still robustly synchronized, and an a priori bound on the approximation error is guaranteed to compare the behaviors of the full-order and reduced-order Lur'e subsystems