Critical Eigenvalues Research Articles

A Rayleigh quotient method for criticality eigenvalue problems in neutron transport

The alpha- and k-effective eigenproblems describe the criticality and fundamental neutron flux mode of a nuclear system. Traditionally, the alpha-eigenvalue problem has been solved using methods that focus on supercritical systems with large, positive eigenvalues. These methods, however, struggle for very subcritical problems where the negative eigenvalue can lead to negative absorption, potentially causing the methods to diverge. We present Rayleigh quotient methods that are applied to demonstrably primitive discretizations of the one-dimensional slab, multigroup in energy, neutron transport equation. These methods are capable of solving subcritical and supercritical alpha- and k-effective eigenvalue problems. The derived eigenvalue updates are optimal in the least squares sense and positive eigenvector updates are guaranteed from the Perron-Frobenius Theorem for primitive matrices.

Study of Rayleigh–Bénard Convection of a Newtonian Nanoliquid in a High Porosity Medium Using Local Thermal Non-equilibrium Model

In the paper we make linear and non-linear stability analyses of Rayleigh–Benard convection in a Newtonian, nanoliquid-saturated porous medium using local thermal non-equilibrium model (LTNE). The LTNE assumption results in advanced onset of convection and increase in heat transport when compared to that of local thermal equilibrium assumption. Free–free and rigid–rigid, isothermal boundaries are considered for investigation. The Galerkin method is used to obtain the critical eigen value. The influence of inter-phase heat transfer coefficient, ratio of thermal conductivities, Brinkman number, porous parameter on the onset of convection as well as on heat transport has been presented graphically and discussed in detail. The effect of increasing the value of porosity modified thermal conductivities advances the onset of convection and enhances the amount of heat transport whereas the remaining parameters have an opposing influence on both onset of convection as well as heat transport.

A Sequential Approach for Small Signal Stability Enhancement With Optimizing Generation Cost

A small signal stability enhancement method is proposed based on optimal power flow model, where small signal stability (SSS) is considered as additional constraint. For handling the discontinuity and high nonlinearity of SSS constraint, a sequential approach is proposed to decompose the original optimization problem into a sequence of sub-problems. For every sub-problem, SSS constraint (i.e., critical eigenvalues) is linearized locally within bounded region. To ensure the effectiveness of eigenvalue sensitivity, a trust region strategy is used for dynamically updating the region size in solving sub-problem sequence. For preventing constrained eigenvalues from varying between critical and uncritical modes in the solving process, an active set strategy is adopted to construct the constraint set and control the solving process. A set of candidate operating points, with different generation cost and minimum damping ratio, is extracted from solution of the sub-problem sequence, which allows to make trade-off between economic efficiency and stability level for system operation. Three test systems are used to validate the effectiveness and applicability of the proposed method.

Numerical solution of two-energy-group neutron noise diffusion problems with fine spatial meshes

The paper presents the development of a strategy for the fine-mesh full-core computation of neutron noise in nuclear reactors. Reactor neutron noise is related to fluctuations of the neutron flux induced by stationary perturbations of the properties of the system. Its monitoring and analysis can provide useful insights in the reactor operations. The model used in the work relies on the neutron diffusion approximation and requires the solution of both the criticality (eigenvalue) and neutron noise equations. A high-resolution spatial discretization of the equations is important for an accurate evaluation of the neutron noise because of the strong gradients that may arise from the perturbations. Considering the size of a nuclear reactor, the application of a fine mesh generates large systems of equations which can be challenging to solve. Then, numerical methods that can provide efficient solutions for these kinds of problems using a reasonable computational effort, are investigated. In particular the power method accelerated with the Chebychev or JFNK-based techniques for the eigenvalue problem, and GMRES with the Symmetric Gauss-Seidel, ILU, SPAI preconditioners for the solution of linear systems, are evaluated with the computation of neutron noise in the case of localized perturbations in 1-D and 2-D simplified reactor cores and in a 3D realistic reactor core.

Stability analysis of anisotropic universe in nonlinear electrodynamics

This paper is devoted to analyze the stability of locally rotationally symmetric Bianchi type I model through phase space portrait in the presence of nonlinear electrodynamics. The normalized dimensionless quantities are introduced to construct an autonomous system of equations. The critical points and respective eigenvalues are evaluated for different linear forms of coupling between scalar field models and dark matter to discuss the stability of the cosmos. We conclude that the increase/decrease of stability in the presence of nonlinear electrodynamics completely depends on the choice of interaction terms between matter and dark energy models.

Enabling Highly Efficient Eigen-Analysis of Large Delayed Cyber-Physical Power Systems by Partial Spectral Discretization

The existing spectral discretization-based methods (SDMs) are capable of accurately computing critical eigenvalues of large power systems when time delays in wide-area damping control loops are considered. However, SDMs suffer from huge dimension of the discretized matrices of spectral operators, which is usually dozens of times of actual system states. To resolve the problem, an idea of partial spectral discretization (PSD) is proposed in this paper where only the retarded state variables instead of all system states are discretized. Following the PSD idea, a partial and explicit infinitesimal generator discretization (PEIGD) method is presented for highly efficient eigen-analysis of large closed-loop delayed cyber-physical power system (DCPPS). In contrast to the original EIGD method, the order of the resultant discretized matrix of the infinitesimal generator is greatly reduced and close to the number of actual state variables. The computational burden of PEIGD can be an order of magnitude less than that of EIGD and nearly the same as eigen-analysis of a system without time delay. Moreover, PEIGD is endowed with exactly the same accuracy as EIGD in capturing critical oscillation modes. Both theoretical analyses and intensive tests on the 16-generator 68-bus test system as well as a 516-bus and a 33028-bus real-life large power systems verify the accuracy and efficiency of the presented method.

Dynamical Stability of Bulk Viscous Isotropic and Homogeneous Universe

In this paper, we study the phase space portrait of homogeneous and isotropic universe by taking different coupling functions between dark energy models and bulk viscous dark matter. The dimensionless quantities are introduced to establish an autonomous set of equations. To analyze the stability of the cosmos, we evaluate critical points and respective eigenvalues for different dynamical quantities. For bulk viscous matter and radiation in tachyon coupled field, these points show stable evolution when γ ≫ δ but accelerated expansion of the universe for δ > 1 9 . The stability of the universe increases for some stationary points which may correspond to the late-time expansion for the coupled phantom field.

Rogue waves generation through multiphase solutions degeneration for the derivative nonlinear Schrödinger equation

The generation of rogue waves from the development of modulational instability of the plane waves due to small perturbations or interactions of solitons is usually modeled by some breather solutions, which are considered to as solitons on a finite background. The instability of small-amplitude perturbations is usually chaotic and may contain many frequencies in their spectra. Thus, a more general description of rogue wave generation can be achieved via the consideration of multiphase solutions and their interactions. The general N-order phase solutions of the derivative nonlinear Schrodinger (DNLS) equation are constructed from the trivial seed (zero solution) by using the determinant representation of the N-fold Darboux transformation. By adjusting the relative phases of the multiphase solutions in the interacting area, namely taking the limit $$\lambda _k \rightarrow \lambda _\mathrm{c}$$ for some of $$\lambda _k$$ (not for all of them), where $$\lambda _\mathrm{c}$$ is a critical eigenvalue associated with the synchronization of each phase in the multiphase nonlinear waves of the DNLS equation, it is possible to obtain different types of quasi-rational solutions from the multiphase solutions degeneration mechanism: quasi-rational traveling wave solutions, rogue waves, and quasi-rogue waves with periodic conditions. Hence, this multiphase solution degeneration represents a new mechanism of rogue wave generation.

Padé-Based Stability Analysis for a Modular Multilevel Converter Considering the Time Delay in the Digital Control System

The digital control system of modular multilevel converters (MMCs) normally synchronizes with the switching actions. The switching frequency of an MMC is generally selected as low as possible from an efficiency point of view, which leads to an unavoidable and relatively long time delay in the control system. Such a time delay might deteriorate the MMC performance and even reduce the stability margin by introducing infinite uncertain eigenvalues into the system state functions. Without a proper time-delay model and quantitative system stability analyses, MMCs might suffer from uncertainties and unstable risks during their operations, especially when the switching frequency is low. This paper derives an MMC model considering the time delay in the digital control system. A Pade-based stability analysis method is then proposed, which consists of three steps: information integration and critical eigenvalue extraction, and trajectory visualization and critical angle (CA) calculation, and quantitative assessment and extreme value determination. By the proposed model and method, the critical eigenvalue can be effectively extracted. With the Hopf bifurcation, the time delay in the digital control system induces a pair critical conjugate eigenvalues into MMC systems, which is detected as an instability cause. The CA and the critical impact are defined to quantitatively assess the stability, while selecting the controller parameters and minimum switching frequency. The proposed method for the MMC stability assessment is experimentally verified.

Estimating Code Biases for Criticality Safety Applications with Few Relevant Benchmarks

Criticality safety analyses rely on the availability of relevant benchmark experiments to determine justifiable margins of subcriticality. When a target application lacks neutronically similar benchmark experiments, validation studies must provide justification to the regulator that the impact of modeling and simulation limitations is well understood for the application and often must provide additional subcritical margin to ensure safe operating conditions. This study estimated the computational bias in the critical eigenvalue for several criticality safety applications supported by only a few relevant benchmark experiments. The accuracy of the following three methods for predicting computational biases was evaluated: the Upper Subcritical Limit STATisticS (USLSTATS) trending analysis method; the Whisper nonparametric method; and TSURFER, which is based on the generalized linear least-squares technique. These methods were also applied to estimate computational biases and recommended upper subcriticality limits for several critical experiments with known biases and for several cases from a blind benchmark study. The methods are evaluated based on both the accuracy of their predicted computation bias and upper subcriticality limit estimates, as well as on the consistency of the methods’ estimates, as the model parameters, covariance data libraries, and set of available benchmark data were varied. Data assimilation methods typically have not been used for criticality safety licensing activities, and this study explores a methodology to address concerns regarding the reliability of such methods in criticality safety bias prediction applications.

Stability analysis of coupled phantom and tachyon field models in nonlinear electrodynamics

This paper is devoted to studying the phase space portrait of FRW universe by taking different linear forms of coupling between scalar field models and dark matter with nonlinear electromagnetic effects. We introduce normalized dimensionless quantities to construct an autonomous system of equations. We evaluate critical points and respective eigenvalues for different parameters to analyze the stability of the cosmos. These points show saddle/unstable behavior for tachyon coupled field with the nonaccelerating universe. In the case of phantom energy, the stability decreases for some critical points which also represent the nonaccelerating universe. We conclude that the dynamical stability of the isotropic and homogeneous universe model reduces in the presence of nonlinear electrodynamics for tachyon as well as the phantom field.

Probabilistic small-signal stability analysis of power system with solar farm integration

Currently, large-scale solar farms are being rapidly integrated in electrical grids all over the world. However, the photovoltaic (PV) output power is highly intermittent in nature and can also be correlated with other solar farms located at different places. Moreover, the increasing PV penetration also results in large solar forecast error and its impact on power system stability should be estimated. The effects of these quantities on small-signal stability are difficult to quantify using deterministic techniques but can be conveniently estimated using probabilistic methods. For this purpose, the authors have developed a method of probabilistic analysis based on combined cumulant and Gram– Charlier expansion technique. The output from the proposed method provides the probability density function and cumulative density function of the real part of the critical eigenvalue, from which information concerning the stability of low-frequency oscillatory dynamics can be inferred. The proposed method gives accurate results in less computation time compared to conventional techniques. The test system is a large modified IEEE 16-machine, 68-bus system, which is a benchmark system to study low-frequency oscillatory dynamics in power systems. The results show that the PV power fluctuation has the potential to cause oscillatory instability. Furthermore, the system is more prone to small-signal instability when the PV farms are correlated as well as when large PV forecast error exists.

Identification of Critical Parameters Affecting Voltage and Angular Stability Considering Load-Renewable Generation Correlations

The renewable energy source based generating technologies and flexible demand and storage devices exhibit significant temporal and spatial uncertainties in generating and loading profiles and introduce additional level of uncertainty in network operation. The dynamic behaviors of such a network can be affected and the stable operation may be compromised. This paper proposes a probabilistic analysis approach for the evaluation of the effect of uncertain parameters on power system voltage and angular stability. Load margin, the damping of critical eigenvalues and the transient stability index have been chosen as the relevant stability indices for voltage stability, small-disturbance stability, and transient stability analysis, respectively. The Morris screening sensitivity analysis method coupled with a multivariate Gaussian copula to account for parameter correlations is used for the priority ranking of uncertain parameters. The approach is illustrated on a number of case studies using modified IEEE 68-bus NETS-NYPS test system. The results obtained in this paper reveal that the critical parameters appear as groups if the input dataset is correlated, and, hence even a parameter (which may be uninfluential individually) can have a significant impact on system dynamic behavior due to its correlation with other influential parameters.

Generic bifurcations in fractional thermo‐mechanics with peridyamic effects

AbstractIn dynamic stability analysis, mathematical aspects of non‐locality are studied by using the theory of dynamical systems. The set of basic equations describing the behavior of continua is transformed to an abstract dynamical system. Such approach results in conditions for cases, when the differential operators have critical eigenvalues of zero real‐parts (dynamic stability or instability conditions). When the critical eigenvalues have nontrivial eigenspace, the way of loss of stability can be classified as a generic bifurcation. Our experiences show that material non‐locality and the generic nature of bifurcation at instability are connected, and the basic functions of the non‐trivial eigenspace can be used to determine internal length quantities of non‐local mechanics. In the paper, non‐local effects are introduced via fractional stress and strain into the equation of motion and into constitutive relations. Then, by defining dynamical systems, stability and bifurcation are studied for states of thermo‐mechanic solids. Conditions for stability and generic bifurcation are presented for constitutive relations under consideration.

Eigen-Analysis of Large Delayed Cyber-Physical Power System by Time Integration-Based Solution Operator Discretization Methods

In this paper, solution operator discretization methods with linear multistep and implicit Runge–Kutta (SOD-LMS/IRK) are presented for efficient eigen-analysis of large delayed cyber-physical power system (DCPPS). First, the time integration-based discretization generates highly structured approximate matrices to the solution operator. Exploitation of the structure ensures low computational burden and high efficiency in solving the matrix-inversion-vector product involved in eigenvalue computation. Second, the implicitly restarted Arnoldi algorithm is employed to compute critical eigenvalue from the solution operator's discretized matrices. SOD-LMS/IRK are endowed with scalability in analyzing very large DCPPS by fully utilizing the inherent sparsity in the augmented system state matrices. The main contribution of the presented SOD-LMS/IRK is the improved efficiency and scalability of existing SOD-PS (pseudo-spectral collocation) and SOD-LMS/IRK. Numerical results on the 16-generator 68-bus test system, a 516-bus, and a 33028-bus real-life large transmission systems validate the effectiveness of the proposed SOD-LMS/IRK.

Complex spatiotemporal behavior and coherent excitations in critically-coupled chains of neural circuits.

We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the "synaptic matrix" embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a "critical" synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period , thus given by a unitary matrix . It is now explicit that such critical couplings are volume-preserving in the sense of Liouville's theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.

Degeneracy in bright–dark solitons of the Derivative Nonlinear Schrödinger equation

Bright–dark soliton interactions modelled by the Derivative Nonlinear Schrödinger (DNLS) equation are constructed from non-vanishing boundary conditions by the N-fold Darboux transformation. Adjusting the limitation λk→λc1(λc2), where λc1(λc2) is a critical eigenvalue associated with the synchronization of the relative phase of the bright–dark solitons in the interacting area, enables to obtain different types of quasi-rational solutions from the bright–dark solitons degeneration. Namely, quasi-rational bright and dark solitons, quasi-rational bright–dark solitons and rogue waves are found. Since a large number of preceding researchers have already addressed the relationship between the breather solutions and rogue waves, a more general modelling of rogue waves can be realized via the consideration of degeneration of bright–dark solitons. Hence, the phenomenon discussed here represents a novel nonlinear mechanism for the generation of rogue waves.

A Parallelized Contour Integral Rayleigh–Ritz Method for Computing Critical Eigenvalues of Large-Scale Power Systems

A contour integral based spectrum transform (CIST) is first introduced for computing critical eigenvalues of power systems. Different from commonly-used shift-invert and Cayley transforms, CIST is able to give equal dominance to the eigenvalue set in a customizable complex plane area, which is defined by a given integral curve. The spectral property of CIST brings high distinctiveness of eigenvalues enclosed by the integral curve in transformed spectrum, along with well convergence of these eigenvalues in subspace methods. With CIST, Rayleigh–Ritz process is used to build subspace and extract eigenvalues. An iterative scheme is proposed for subspace refinement. The iterative contour integral Rayleigh–Ritz method is implemented with full utilization of parallel potentiality in contour integral numerical evaluation. Experiments on three test systems of different scales are performed to validate the reliability, computational efficiency, and parallel scalability of the proposed method.

Stability Robustness for Secondary Voltage Control in Autonomous Microgrids With Consideration of Communication Delays

The future extensive usage of open communication networks for secondary voltage control in autonomous microgrids (MGs) inherently introduce time delays, which would degrade the system performance or even cause instability in the worst case. However, existing literatures have largely ignored these communication delays between the microgrid central controller and primary local controllers. To fill the knowledge gap, this paper proposes an analytical method to determine the stability robustness of secondary voltage control in MGs by taking into account the communication latency effect. Based on static output feedback, a closed loop small-signal model of an MG installed with proportional-integral secondary voltage controllers is first derived. Delay-dependent criteria are then formulated to detect the stability posture of MG system by tracing critical eigenvalues and cluster treatment of characteristic roots, in the forms of delay margin for a single delay and stability tableau over the delay domain for multiple delays, respectively. By a series of trial declarations, the qualitative impact of secondary controller gains on system stability regions against uncertain delays is investigated, from which it can be observed that generally the increase of integral gains would yield to a less delay-dependent stability of control system and the results can be utilized to guide the controller design for a compromise between the dynamic performance and stability robustness. The effectiveness of the proposed methodology is verified by a simulation study.

Traveling Waves Bifurcating from Plane Poiseuille Flow of the Compressible Navier–Stokes Equation

Plane Poiseuille flow in viscous compressible fluid is known to be asymptotically stable if Reynolds number R and Mach number M are sufficiently small. On the other hand, for R and M being not necessarily small, an instability criterion for plane Poiseuille flow is known, and the criterion says that, when R increases, a pair of complex conjugate eigenvalues of the linearized operator cross the imaginary axis. In this paper it is proved that a spatially periodic traveling wave bifurcates from plane Poiseuille flow when the critical eigenvalues cross the imaginary axis.