Equivalence Of Dynamical Systems Research Articles

Online Application of Local OOS Protection and Graph Theory for Controlled Islanding

Despite layers of protection at work, power systems have experienced major blackouts in the past due to cascading outages. Such outages can be prevented with controlled islanding, which consists of two problems: “when” and “where” to island. The majority of the existing offline and real-time methods for the “when” aspects are based on wide area measurement systems (WAMSs), which depend on the system topology to perform periodic offline simulations or real-time computation of dynamic system equivalents. In this paper, a simplified “when” approach is proposed based on local generator out-of-step (OOS) protection and generator coherency using fault location. The novelty of the work lies in the use of local OOS protection relays to identify the overall system instability, which eliminates the need for WAMS communication and the complex computation required in real-time methods to form dynamic system equivalents. As this approach does not require any equivalent networks, the methodology is independent of system topology. Unlike existing WAMS-based methods, the proposed methodology simplifies the communication between the central “when” unit and each generator’s OOS protection relay using status flags communicated with the IEC 61850 R-GOOSE protocol. The proposed “when” methodology is combined with the “where” method based on graph theory to test the overall controlled islanding scheme in the IEEE 39-bus system using the real-time digital simulator (RTDS).

Application of Balanced Realizations for Model-Order Reduction of Dynamic Power System Equivalents

This paper reviews and applies the balanced realization (BR) theory to obtain reduced-order models from dynamic system equivalents of electric power networks. BR allows obtaining reduced-order models via a process in which the original asymptotic stable system is internally balanced. The balanced system is truncated according to its dominant dynamics. It can be proven that the truncated, that is, reduced-order, system is also stable. This paper applies the BR method to dynamic system equivalents represented as frequency-dependent network equivalents (FDNEs). Furthermore, it shows that an assumed reduced-order FDNE can be further reduced via the BR process. Four case studies, involving one transmission network and three wind power plants, are presented.

Dynamic System Equivalents: A Survey of Available Techniques

This paper presents a brief review of techniques available for reducing large systems to smaller equivalents. This paper is divided into high frequency equivalents, low frequency equivalents, and wideband equivalents. Numerical examples are presented to demonstrate selected methods of high frequency equivalencing.

Dynamic System Equivalents: A Survey of Available Techniques

Dynamic System Equivalents: A Survey of Available Techniques

$\mathrm{C}^*$-Algebras

A C\*-algebra is an involutive Banach algebra $A$ which satisfies the C\*-condition $|a^\*a|=|a|^2$ for all $a\in A$. The theory of C\*-algebras goes back to work of Murray and von Neumann, who first studied a special variant now known as von Neumann algebras. The theory developed rapidly after some ground breaking work of Gelfand and Naimark in 1943 in which they showed that \begin{itemize} \item every commutative C\*-algebra can be realized up to isomorphism as an algebra of continuous functions which vanish at infinity on a locally compact Hausdorff space, \item every C\*-algebra can be realized as a closed \*-subalgebra of the algebra of bounded operators $B(\mathcal H)$ on some Hilbert space $\mathcal H$. \end{itemize} In the 70's and 80's of the last century, the first of the above items lead to the point of view that non-commutative C\*-algebras should be regarded as function spaces of \`\`non-commutative'' topological spaces. As a consequence, completely new areas in mathematics, like Non-commutative Geometry or Free Probability evolved and we now see that the theory of C\*-algebras became a very active field with applications in and interactions with almost all areas of modern mathematics. The aim of the workshop \emph{C\*-algebras}, organized by Claire Anantharaman-Delaroche, Siegfried Echterhoff, Uffe Haagerup, and Dan Voiculescu, is to bring together leading researchers from basically all areas related to the field. This gives a unique opportunity to maintain a broad view on the subject and to create new cooperations between researchers with different background. Among the 42 participants was a good number of young researchers, some of them already on the top of the field. There have been 27 lectures presented at the workshop with topics ranging from classification of C\*-algebras, group actions on C\*-algebras, orbit equivalence of dynamical systems, Jones' theory of subfactors, fundamental groups of II$\_1$-factors, $L^2$-invariants, duality for quantum groups and quantum groupoids, Index theorems, C\*-algebras related to number theory, free probability, the relation between C\*-algebras and Harmonic Analysis, and others. Among the most exciting recent developments in the field we mention the breathtaking advances of Popa and Vaes on the fundamental groups of II$\_1$-factors in the theory of von Neumann algebras, with beautiful applications to the classification of equivalence relations and dynamical systems. Other milestones are the progress of Kirchberg and coworkers on the classification of actions on $\mathcal O\_2$ and the beautiful results of Dadarlat on continuous bundles of C\*-algebras. Important progress has been achieved also in the classification theory of simple and non-simple C\*-algebras by Elliot, R{\o}rdam, Toms, and Winter--with some further progress being made during the workshop. There were many other exciting contributions to this very successful workshop, which can be seen from the reports presented in this issue. It is a pleasure for the organizers to thank all participants of the workshop for their beautiful lectures and fruitful discussions. We also want to use this opportunity to thank the Mathematisches Forschungs\\-institut Oberwolfach for providing a very stimulating environment and strong support for organizing this conference. Special thanks also go to the very competent and helpful staff of the institute.

Equivalence of Dynamical Systems by Bisimulation

A general notion of bisimulation is defined for linear input-state-output systems, using analogies with the theory of concurrent processes. A characterization of bisimulation and an algorithm for computing the maximal bisimulation relation is derived using geometric control theory. Bisimulation is shown to be a notion which unifies the concepts of state-space equivalence and state-space reduction, and which allows to study equivalence of systems with nonminimal state-space dimension. The notion of bisimulation is especially powerful for "nondeterministic" dynamical systems, and leads in this case to a notion of equivalence which is finer than equality of external behavior. For abstractions of systems it is shown how the results specialize to previously obtained results by other authors. Extensions of the main results to the nonlinear case are provided.

GENERALIZED SYNCHRONIZATION, FREQUENCY-LOCKING AND PHASE-LOCKING OF COUPLED SINE CIRCLE MAPS

Applying invariant manifold theorem, we study the existence of generalized synchronization of a coupled system, with local systems being different sine circle maps. We specify a range of parameters for which the coupled system achieves generalized synchronization. We also investigate the relation between generalized synchronization, predictability and equivalence of dynamical systems. If the parameters are restricted in the specified range, then all the subsystems are topologically equivalent, and each subsystem is predictable from any other subsystem. Moreover, these subsystems are frequency locked even if the frequencies are greatly different in the absence of coupling. If the local systems are identical without coupling, then the widths of the phase-locked intervals of the coupled system are the same as those of the individual map and are independent of the coupling strength.

New extended spectral decompositions of the Renyi map

We obtain two new spectral decompositions of the Renyi map. The first has eigenvalues 3 − n and is associated with the Cantor type observables. The second has eigenvalues 2 − n and is associated with analytic observables with singularities at the boundary. We discuss the natural equivalence of dynamical systems with respect to the approach to equilibrium.

Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems

Necessary and sufficient conditions for the occurrence of generalized synchronization of unidirectionally coupled dynamical systems are given in terms of asymptotic stability. The relation between generalized synchronization, predictability, and equivalence of dynamical systems is discussed. All theoretical results are illustrated by analytical and numerical examples. In particular, the existence of generalized synchronization in the case of parameter mismatch between coupled systems leads to a new interpretation of recent experimental results. Furthermore, the possible application of generalized synchronization for attractor reconstruction in nonlinear time series analysis is discussed.

Topological entropy and equivalence of dynamical systems

Topological entropy and equivalence of dynamical systems

Asymptotic equivalence of dynamical systems

Asymptotic equivalence of dynamical systems

Stability, attraction properties and asymptotic equivalence of dynamical systems

Stability, attraction properties and asymptotic equivalence of dynamical systems