Topological Conditions Research Articles

Gap sequences and Topological properties of Bedford–McMullen sets* *The research of Miao is partially supported by Shanghai Key Laboratory of PMMP (18dz2271000) and NSFC Grant 12071137. The research of Ruan is partially supported by NSFC Grant 11771391, ZJNSF Grant LY22A010023 and the Fundamental Research Funds for the Central Universities of China Grant 2021FZZX001-01.

In this paper, we study the topological properties and the gap sequences of Bedford–McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford–McMullen sets and prove that all normal Bedford–McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedford–McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence.

An Advanced and Robust Approach to Maximize Solar Photovoltaic Power Production

The stochastic and erratic behavior of solar photovoltaic (SPV) is a challenge, especially due to changing meteorological conditions. During a partially irradiated SPV system, the performance of traditional maximum power point tracking (MPPT) controllers is unsatisfactory because of multiple peaks in the Power-Voltage curve. This work is an attempt to understand the performance uncertainties of the SPV system under different shading conditions and its mitigation. Here, a novel hybrid metaheuristic algorithm is proposed for the effective and efficient tracking of power. The algorithm is inspired by the movement of grey wolves and the swarming action of birds, and is thus known as the hybrid grey wolf optimizer (HGWO). The study focuses on the transient and steady-state performance of the proposed controller during different conditions. A comparative analysis of the proposed technique with incremental conductance and a particle swarm optimizer for different configurations is presented. Thus, the results are presented based on power extracted, shading loss, convergence factor and efficiency. The proposed HGWO–MPPT is found to be better as it has a maximum efficiency of 94.30% and a minimum convergence factor of 0.20 when compared with other techniques under varying conditions for different topologies. Furthermore, a practical assessment of the proposed controller on a 6.3 kWp rooftop SPV system is also presented in the paper. Energy production is increased by 8.55% using the proposed approach to the practical system.

Travel Time Reliability of Highway Network under Multiple Failure Modes

Network reliability reflects a system’s ability to perform specified functions under specified topological and traffic conditions. Network reliability is the weighted sum of connection reliability and travel time reliability. Based on complex network theory, a new method was proposed to calculate the travel time reliability of road networks. The topology model of a regional highway network in China was built using the dual method. After a random attack or deliberate attack, node sizes in the sub-network can be used to reflect the node importance for network connection reliability. Some conclusions were drawn after the change in travel time coefficient and delay coefficient. The increases in the two coefficients will accelerate the decrease of travel time reliability of the highway network. After a comparison among three methods of travel time reliability, including variation coefficient, the misery indexes, and the new equation, the new method was further verified. The influence factors of highway network reliability were analyzed under the condition of different highway blockage and congestion.

Assessment of the voltage stability index of electric networks supplying charging stations using a multilayer perceptron

Ensuring power system voltage stability is one of the key challenges in power system planning and operation. At the moment, as a result of a number of technological incidents in the form of a violation of the stability of the voltage of the power system, multilayer perceptrons were tested on the territory of various countries.PURPOSE. Development of an intelligent approach to assessing the Load Stability Index (LSI) in the power system using computational intelligence methods (neural networks).METHODS. In the course of these studies, a method for estimating the ISI was used, which is used to monitor the stability of voltage in the power system using the "Smart Park" calculation method.RESULTS. The results of providing the development stage are obtained, which substantiate the possibility of using the Multilayer Perceptron (MLP) for estimating the IEL with a high degree of accuracy.CONCLUSION. In the course of these studies, the authors of the article put forward the following conclusions: an approach based on a multilayer perceptron (MPP) neural network with feed-forward is presented; an assessment of the ISI was carried out on the example of a power system using the calculation method "Smart Park"; the presented neural network approach does not depend on the open-circuit voltage on a particular load bus; the presence of indicators of active power, reactive power and voltage on the load bus is sufficient to assess the IPL; The MSP approach allows for an accurate assessment of the ISI even when the topology and operating conditions change. The obtained indicators can be applied in changing the methodology for assessing the stability of voltage in electrical systems and complexes and calculating their operating modes.

Triangular angle rigidity for distributed localization in 2D

Recent advances in sensing technology have enabled sensor nodes to measure interior angles with respect to their neighboring nodes. However, it is unknown which combination of angle measurements is necessary to make a sensor network localizable, and it is also unidentified if there is a distributed localization algorithm whose required communication only consists of the sensor nodes’ measured angles and estimated positions. Motivated by these two challenging problems, this paper develops triangular angle rigidity for those networks consisting of a set of nodes and triangular angle constraints in 2D. First, we transfer the geometric constraint of each triangle into an angle-induced linear constraint. Based on the linear constraint, we show that different from angle rigidity, triangular angle rigidity implies global triangular angle rigidity. More importantly, inspired by Laman’s theorem, we propose a topological, necessary and sufficient condition to check generic triangular angle rigidity. Based on the results on triangular angle rigidity, both algebraic and topological localizability conditions are developed, which are necessary and sufficient when the number of anchor nodes in the network is two. Both continuous and discrete localization algorithms are proposed, in which only measured angles and estimated positions are communicated among the sensor nodes. Finally, a simulation example with 32 sensor nodes is used to validate the effectiveness of the proposed approaches.

Fast Low Frequency Fault Location and Section Identification Scheme for VSC-Based Multi-Terminal HVDC Systems

Penetration of DC networks is rapidly increasing in modern power systems. As transient behavior of DC grids is different from that of AC ones, AC grid protection schemes might not be applicable to protect DC networks. Exact fault location is vital for protection of DC networks to reduce repair cost and achieve fast power recovery. This paper proposes a novel topology independent multi-terminal DC scheme to identify fault section and location for DC transmission systems. Voltage and current signals measured at the terminals are analyzed by the least squares error (LSE) algorithm. Then, the proposed scheme uses low frequency components of the signals to locate DC faults offline. Unlike some other methods, it does not require sophisticated sensors and hardware to capture very high frequency content of the signals. Hence, the proposed scheme can be implemented in practice without additional measuring infrastructure. The performance of the proposed scheme is evaluated under various fault conditions for different topologies of DC networks in both overhead lines and cables. Numerous simulations carried out demonstrate that the proposed scheme is quite accurate and provides excellent performance under different fault resistances, sections, and locations.

Distributed Differential Games for Control of Multi-Agent Systems

Motivated by the challenges arising in the field of multi-agent system (MAS) control, we consider linear heterogeneous MAS subject to local communication and investigate the problem of designing distributed controllers for such systems. We provide a game theoretic framework for systematically designing distributed controllers, taking into account individual objectives of the agents and their possibly incomplete knowledge of the MAS. Linear state-feedback control laws are obtained via the introduction of a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributed differential game</i> , namely, the combination of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local non-cooperative differential games</i> , which are solved in a decentralized fashion. Conditions for stability of the MAS are provided for the special cases of acyclic and strongly connected communication graph topologies. These results are then exploited to provide stability conditions for general graph topologies. The proposed framework is demonstrated on a tracking synchronization problem associated with the design of a distributed secondary voltage controller for microgrids and on a numerical example.

Real-Time Transient Instability Detection in the Power System With High DFIG-Wind Turbine Penetration via Transient Energy

By increasing the penetration of wind power in the modern power systems, to investigate transient stability is of special importance. This article presents an innovative index based on the concept of the potential energy boundary surface along with kinetic energy, which aims to detect the transient instability of a power system comprising doubly fed induction generator (DFIG)-based wind farms. Accordingly, transient instability detection (TID) is performed without directly calculating the unstable equilibrium point and thus the computational load is decreased. Since the proposed approach requires only the postfault data that can be easily measured by the phasor measurement units, it is suitable for real-time applications. Therefore, the proposed approach can be applied as a general tool to any power system with any change in topology and operating conditions. Moreover, considering that the current-balance form is the preferred industry model for the implementation of transient stability simulation, the network equations in the current-balance form are extracted for grid-connected DFIG and synchronous generators. Different scenarios are simulated in the Western System Coordinating Council 3-machine, 9-bus system, and the 10-generator New England system. To validate the new index for TID, simulation results are compared with the transient stability index and out-of-step distance relay. The obtained results validate the correctness and effectiveness of the presented new index.

Synchronous and asynchronous resilient impulsive control for group consensus of second-order multi-agent systems with communication delays

This paper studies the resilient group consensus of continuous-time second-order multi-agent systems (MASs) with malicious agents. Adopting the idea that each normal agent ignores the most extreme values from neighbors, synchronous resilient impulsive algorithm based on sampled data is proposed for normal agents with bounded communication delays to achieve group consensus. Meanwhile, asynchronous resilient impulsive algorithm is also proposed for MASs where each agent has its own time clock. Sufficient topological conditions are obtained for solving resilient group consensus under synchronous and asynchronous settings, respectively. Numerical examples are provided to illustrate the effectiveness of the theoretical results.

Almost paracomplex structures on 4-manifolds

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics of any real 3-dimensional space form admits both isometric and anti-isometric paracomplex structures.This paper considers the existence or otherwise of isometric and anti-isometric almost paracomplex structures j on a pseudo-Riemannian 4-manifold (M,g), such that j is parallel with respect to the Levi-Civita connection of g. It is shown that if an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, then the scalar curvature of the metric must be zero. In addition, it is found that j is parallel iff the eigenplanes are tangent to a pair of mutually orthogonal foliations by totally geodesic surfaces.The composition of a Riemannian metric with an isometric almost paracomplex structure j yields a neutral metric g′. It is proven that if j is parallel, then g is Einstein iff g′ is conformally flat and scalar flat.The vanishing of the Hirzebruch signature is found to be a necessary topological condition for a closed 4-manifold to admit an Einstein metric with a parallel isometric paracomplex structure. Thus, while the K3 manifold admits an Einstein metric with an isometric paracomplex structure, it cannot be parallel. The same holds true for certain connected sums of complex projective 2-space and its conjugate.

MONA—A magnetic oriented nodal analysis for electric circuits

AbstractThe modified nodal analysis (MNA) is probably the most widely used formulation for the modeling and simulation of electric circuits. Its conventional form uses electric node potentials and currents across inductors and voltage sources as unknowns, thus taking an electric viewpoint. In this paper, we propose an alternative magnetic oriented nodal analysis (MONA) for electric circuits, which is based on magnetic node potentials and charges across capacitors and voltage sources as the primary degrees of freedom, thus giving direct access to these quantities. The resulting system has the structure of a generalized gradient system which immediately ensures passivity in the absence of sources. A complete index analysis is presented showing regularity of the magnetic oriented formulation under standard topological conditions on the network interconnection. In comparison to conventional MNA, the differential‐algebraic index of MONA is smaller by one in most cases which facilitates the numerical solution. Some preliminary numerical experiments are presented for illustration of the feasibility and stability of the new approach.

A consensus protocol considering Lipschitz constant and communication topology condition of high-order nonlinear multi-agent systems

A new protocol is proposed for the consensus of multi-agent systems with the properties of high-order and nonlinearity for every single system. It is assumed that the nonlinear function of the sub-agent is restricted by the Lipschitz condition, and in addition, there are two kinds of communication topology structures among the members. The protocol combines the eigenvalues of the Laplace matrix and the Lipschitz constant of the multi-agent system. By reasonably designing the Lyapunov function, the convergence condition of the system is divided into two parts which are connected by an unknown quantity k. Except for quantity k, the connection of other parts of the protocol is loose, which makes the protocol more flexible.

Event-Triggered Distributed Estimation with Decaying Communication Rate

We study distributed estimation of a high-dimensional static parameter vector through a group of sensors whose communication network is modeled by a fixed directed graph. Different from existing time-triggered communication schemes, an event-triggered asynchronous scheme is investigated in order to reduce communication while preserving estimation convergence. A distributed estimation algorithm with a single step size is first proposed based on an event-triggered communication scheme with a time-dependent decaying threshold. With the event-triggered scheme, each sensor sends its estimate to neighbor sensors only when the difference between the current estimate and the last sent-out estimate is larger than the triggering threshold. We prove that the proposed algorithm has mean-square and almost-sure convergence respectively, under an integrated condition of sensor network topology and sensor measurement matrices. The condition is satisfied if the topology is a balanced digraph containing a spanning tree and the system is collectively observable. Moreover, we provide estimates for the convergence rates, which are related to the step size as well as the triggering threshold. Furthermore, as an essential metric of sensor communication intensity in the event-triggered distributed algorithms, the communication rate is proved to decay to zero with a certain speed almost surely as time goes to infinity. We show that given the step size, adjusting the decay speed of the triggering threshold can lead to a tradeoff between the convergence rate of the estimation error and the decay speed of the communication rate. Specifically, increasing the decay speed of the threshold would make the communication rate decay faster, but reduce the convergence rate of the estimation error. Numerical simulations are provided to illustrate the developed results.

An Enhanced Differential Protection Scheme for LVDC Microgrid

The low-voltage dc (LVDC) microgrid possesses numerous benefits and their penetration in the power system has increased rapidly in recent years. However, the detection of faults in the LVDC microgrid is a challenging issue due to the large magnitude of fault currents and fault-level variation in the microgrid. The performance of the recent current and its derivative-based protection scheme is limited in case of faults in the islanding mode of operation, different microgrid topologies, varying distributed generations (DGs) penetration, and against the measurement noise. This article presents an enhanced differential protection scheme for LVDC microgrid integrated with multiple DGs and storage. The differential current and its first derivative are processed through the decision tree (DT) algorithm for fault detection and the K-nearest neighbor (KNN) technique is utilized for fault classification. The robustness of the proposed protection scheme is tested for different fault types and fault conditions with variation in microgrid topology and operating conditions. The impact of the intermittent and volatile nature of the DGs, the presence of measurement noise, and assessment during external faults and critical no-fault cases have been investigated. The proposed scheme is tested on the MATLAB/ SIMULINK and validated on the Typhoon HIL platform for the assessment of real-time performance. The test results show that the proposed scheme can detect and classify faults with high accuracy and faster response time and, thus, can be a potential candidate for providing dependable protection measures for LVDC microgrids.

Some classes of topological spaces related to zero-sets

An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briefly CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z#-embedded subspace of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clTZ is a zero-set in T). In 6P.5 of [8] it was shown that a closed countable union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. cozero complemented)-space and other classes of topological spaces near to them. Some algebraic and topological equivalent conditions of CZ (resp. cozero complemented)-space are characterized. Examples are provided to illustrate and delimit our results.

Consensus of cooperative–antagonistic multi-agent networks with asynchronous three-option decision mechanism

Instead of using the conventional discrete-time or continuous-time protocols with arbitrary values, this paper studies a cooperative–antagonistic multi-agent network with three-option decision mechanism. In particular, each agent over the network periodically makes decisions just from three available options (i.e., “1”, “0” and “−1”) at decision instants for the promotion of agreement with its cooperative neighbors or disagreement with its antagonistic neighbors. The asynchronous decisions of distinct agents are allowed herein. A digraph with signs is implemented to characterize the information delivery. Next, the consensus behaviors of the concerned cooperative–antagonistic multi-agent network are analyzed. In particular, on the basis of an intermediate modulus consensus result in terms of a strongly connected topology condition, it is first shown that structural balance is a sufficient and necessary condition for the bipartite consensus result. Besides this, a stabilizing result that all the agents reach an agreement at the origin is shown under the structurally unbalanced condition. On the other hand, for a relaxed signed digraph that contains a spanning tree, if only the root subgraph is strongly connected and structurally balanced, it is shown that only the root agents reach the bipartite consensus while the states of the non-root agents converge between the polarized values of the root agents. In addition, if the structural balance condition associated with the root subgraph cannot be guaranteed, the considered cooperative–antagonistic multi-agent network is shown stabilizing. Finally, simulations are carried out to verify the proposed theoretical results.

Critical parameters of the synchronisation's stability for coupled maps in regular graphs

Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters – coupling strength, maximum Lyapunov exponent, and link density – that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: k-cycles (i.e., regular lattices of even degree k, which can be embedded in Tk tori) and k-Möbius ladders, which we introduce here to generalise the Möbius ladder of degree k = 3. Our results highlight differences in the synchronisation manifold's stability of these graphs – even for identical node degrees – in the finite size and infinite size limit.

Diffeomorphic registration for retinotopic maps of multiple visual regions.

Retinotopic map, the mapping between visual inputs on the retina and neuronal responses on the cortical surface, is one of the central topics in vision science. Typically, human retinotopic maps are constructed by analyzing functional magnetic resonance responses to designed visual stimuli on the cortical surface. Although it is widely used in visual neuroscience, retinotopic maps are limited by the signal-to-noise ratio and spatial resolution of fMRI. One promising approach to improve the quality of retinotopic maps is to register individual subject's retinotopic maps to a retinotopic template. However, none of the existing retinotopic registration methods has explicitly quantified the diffeomorphic condition, that is, retinotopic mapsshall be aligned by stretching/compressing without tearing up the cortical surface. Here, we developed Diffeomorphic Registration for Retinotopic Maps (DRRM) to simultaneously align retinotopic maps in multiple visual regions under the diffeomorphic condition. Specifically, we used the Beltrami coefficient to model the diffeomorphic condition and performed surface registration based on retinotopic coordinates. The overall framework preserves the topological condition defined in the template. We further developed a unique evaluation protocol and compared the performance of the new method with several existing registration methods on both synthetic and real datasets. The results showed that DRRM is superior to the existing methods in achieving diffeomorphic registration in synthetic and empirical data from 3T and 7T MRI systems. DRRM may improve the interpretation of low-quality retinotopic maps and facilitate applications of retinotopic maps in clinical settings.

Dynamic Multicommodity Contraflow Problem with Asymmetric Transit Times

A maximum dynamic multicommodity flow problem concerns with the transportation of several different commodities through the specific source-sink path of an underlying capacity network with the objective of maximizing the sum of commodity flows within a given time horizon. Motivated by the uneven road condition of transportation network topology, we introduce the dynamic multicommodity contraflow problem with asymmetric transit times on arcs that increase the outbound lane capacities by reverting the orientation of lanes towards the demand nodes. Moreover, a pseudo-polynomial time algorithm by using a time-expanded graph and an FPTAS by using aΔ-condensed time-expanded network are presented.

Ricci flow smoothing for locally collapsing manifolds

We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.