Eigenvalue-based Stability Analysis Research Articles

Nonlinear classifiers for wet-neuromorphic computing using gene regulatory neural network

The gene regulatory network (GRN) of biological cells governs a number of key functionalities that enable them to adapt and survive through different environmental conditions. Close observation of the GRN shows that the structure and operational principles resemble an artificial neural network (ANN), which can pave the way for the development of wet-neuromorphic computing systems. Genes are integrated into gene-perceptrons with transcription factors (TFs) as input, where the TF concentration relative to half-maximal RNA concentration and gene product copy number influences transcription and translation via weighted multiplication before undergoing a nonlinear activation function. This process yields protein concentration as the output, effectively turning the entire GRN into a gene regulatory neural network (GRNN). In this paper, we establish nonlinear classifiers for molecular machine learning using the inherent sigmoidal nonlinear behavior of gene expression. The eigenvalue-based stability analysis, tailored to system parameters, confirms maximum-stable concentration levels, minimizing concentration fluctuations and computational errors. Given the significance of the stabilization phase in GRNN computing and the dynamic nature of the GRN, alongside potential changes in system parameters, we utilize the Lyapunov stability theorem for temporal stability analysis. Based on this GRN-to-GRNN mapping and stability analysis, three classifiers are developed utilizing two generic multilayer sub-GRNNs and a sub-GRNN extracted from the Escherichia coli GRN. Our findings also reveal the adaptability of different sub-GRNNs to suit different application requirements.

Stability Analysis in Multi-VSC (Voltage Source Converter) Systems of Wind Turbines

In this paper, a holistic nonlinear state-space model of a system with multiple converters is developed, where the converters correspond to the wind turbines in a wind farm and are equipped with grid-following control. A novel generalized methodology is developed, based on the number of the system’s converters, to compute the equilibrium points around which the model is linearized. This is a more solid approach compared with selecting operating points for linearizing the model or utilizing EMT simulation tools to estimate the system’s steady state. The dynamics of both the inner and outer control loops of the power converters are included, as well as the dynamics of the electrical elements of the system and the digital time delay, in order to study the dynamic issues in both high- and low-frequency ranges. The system’s stability is assessed through an eigenvalue-based stability analysis. A participation factor analysis is also used to give an insight into the interactions caused by the control topology of the converters. Time domain simulations and the corresponding frequency analysis are performed in order to validate the model for all the control interactions under study.

Simplified Virtual Synchronous Compensator With Grid–Forming Capability

The Simplified Virtual Synchronous Compensator (S–VSC) is a Virtual Synchronous Machine (VSM) solution avail- able in the literature, which operates as a virtual compensator. Previous works demonstrated that it can provide grid services (e.g., inertial behavior, grid support during faults) only in grid– following operation. However, the grid–forming capability of the S–VSC (and in general of a virtual compensator) has never been investigated in the literature. Therefore, this paper proposes an S–VSC model able to operate both in grid–following and in grid– forming configuration. First, the paper validates the small–signal stability of the control algorithm both in grid–mode and island operation through an eigenvalue–based stability analysis. Next, the control algorithm is experimentally validated on a 15 kVA inverter connected to a scaled microgrid. The experimental tests demonstrate that the S–VSC can operate both as grid–following and grid–forming. Moreover, it can seamlessly switch from the grid–connected to the island operation without requiring any communication system. Finally, the grid–forming capability of the S–VSC is experimentally validated even in case of a fault occurrence, thus representing a valid solution for the control of a microgrid.

Stability Constrained Optimal Operation of Standalone DC Microgrids Considering Load and Solar PV Uncertainties

Operation and control dynamics of islanded DC microgrids (DC mGs) have received enormous interest recently in distribution system management. In this regard, research concern on droop-controlled distributed generations (DGs) integrated within the standalone DC mG is becoming more profound. This paper proposes an optimal operation of droop-controlled standalone DC mG with combined small-signal stability and economic-environment-related objectives. The proposed approach optimizes the principal eigenvalue that affects the system stability and two other objectives. An eigen value-based stability analysis for network configurations, with the possibility of its buses solely connected to either dispatchable DG or constant power load (CPL), is proposed. Moreover, uncertainties involved in mG network variables that originate due to the integration of solar PV generation with load consumption were considered and modelled with relevant probability characterization. A recent swarm-based intelligent technique (dragonfly algorithm) for optimization of droop parameters is adopted in the proposed approach. A unique bi/tri-objective combinations are solved, and obtained results are discussed. To show the adaptability of the proposed approach for any given DC mG network, a standalone 6-bus mG was adopted to verify the applicability. Furthermore, the time-domain simulations were carried out on a sample 3-bus DC mG to validate the proposed strategy.

Stability Impacts of an Alternate Voltage Controller (AVC) on Wind Turbines with Different Grid Strengths

This paper studies the stability impact of the alternate voltage controller’s (AVC) low-pass filter (LPF) in a wind turbine’s grid-connected voltage source converter (VSC). A small-signal model of the grid-connected converter is designed with a grid-following synchronization control. More specifically, the non-linear state-space model of the grid-connected converter was developed, including the dynamics of both the inner and outer control loops of the converter, the dynamics of the elements of the electrical system, as well as the digital time delay. An eigenvalue-based stability analysis gives insight into the stability impacts of the outer-loop controllers. It is proven that the cutoff frequency of the AVC’s LPF affects the phase-locked loop (PLL) and AVC bandwidths of instability, as well as the corresponding critical oscillation frequencies. This phenomenon is observed in both weak and strong grids. Consequently, the small-signal stability regions of the PLL and AVC bandwidth can be identified for the range of the AVC’s LPF cutoff frequency under study. The stability regions of the PLL and AVC, which are obtained from the small-signal model, as well as the determined critical oscillation frequencies, are validated through time domain simulations and fast-Fourier transformation (FFT) analysis.

Comparative Stability Analysis and Improvement of Grid-Following Converters Using Novel Interpretation of Linear Time-Periodic Theory

This article presents a comparative stability investigation of grid-following (GFL) converters with different advanced phase-locked loops (PLLs) and current control schemes based on linear time-periodic (LTP) theory. First, a time-domain physical interpretation of the LTP theory is proposed, which inspires an iterative LTP eigenvalue computation algorithm. A measure for the influence of the frequency coupling effect on each eigenvalue is defined to demonstrate the necessity of applying the LTP theory. Then, eigenvalue-based stability analysis is carried out to gain insights into dynamic characteristics of two implementations of GFL converters, namely, dual synchronous reference frame PLL (DSRF-PLL) and proportional integral (PI) current control, and dual second-order generalized integrator PLL (DSOGI-PLL) and proportional resonant (PR) current control. The effects of the PLL bandwidth, the current control time constant, the grid strength, and imbalance on the system stability are evaluated. In addition, a damping loop is proposed to improve the stability margin of GFL converters connected to weak grids. Finally, MATLAB/Simulink simulations and experimental measurements validate the correctness of the proposed modeling method and stability assessment results.

Induced aseismic slip and the onset of seismicity in displaced faults

Abstract We address aseismic fault slip and the onset of seismicity resulting from depletion-induced or injection-induced stresses in reservoirs with pre-existing vertical or inclined faults. Building on classic results, we discuss semi-analytical modelling techniques for fault slip including dislocation theory, Cauchy-type singular integral equations and the use of Chebyshev polynomials for their solution and an eigenvalue-based stability analysis. We consider slip patch development during depletion for faults with zero, constant static and slip-weakening friction, and our results confirm earlier findings based on numerical simulation, in particular the aseismic growth of two slip patches that may subsequently merge and/or become unstable resulting in nucleation of seismic slip. New findings include improved approximate expressions for the induced seismic moment per unit strike length and a description of the effect of coupling between the slip patches which affects both forward simulation and eigenvalue computation for high values of the ratio of fault throw to reservoir height. Our implementation based on analytical inversion and semi-analytical integration with Chebyshev polynomials is more efficient and more robust than our numerical integration approach. It is not yet well suited for Monte Carlo simulation, which typically requires sub-second simulation times, but with some further development that option seems to be within reach. Moreover, our results offer a possibility for embedded fault modelling in large-scale numerical simulation tools.

State Space Modeling of an Offshore Wind Power Plant With an MMC-HVDC Connection for an Eigenvalue-Based Stability Analysis

Large offshore wind power plants (OWPPs) installed far from the coastline are emerging to benefit from the strong and steady wind resources available at these locations. The high-voltage direct-current (HVDC) transmission system based on the modular multilevel converter (MMC) is the most appropriate solution to transmit the produced energy to the onshore grid, in a way that a complex power-electronic-based electrical system is formed at the OWPP. Undesired interactions can occur between the MMC-based HVDC station, the several wind-turbine (WT) converters, and the passive elements of the offshore grid. To guarantee a safe and reliable operation of the OWPP, a small-signal analysis must be performed in advance to predict possible unstable situations and their root causes, in a way as to take corrective measures to avoid them. In this paper, a state space model of an OWPP is developed in which a recently proposed model in multiple <i>dq</i> frames is adopted for the MMC. With the developed model, an eigenvalue-based stability assessment is carried out, where participation factors are calculated to identify the main contributors to the unstable modes that appear as a consequence of undesired low-frequency interactions at the offshore grid. As far as the authors know, an eigenvalue-based stability assessment of an OWPP with an MMC-HVDC connection is a topic rarely or never explored in the literature before and, thus, this is the main contribution of this paper. An adaptation to the MMC state space model is proposed so that the converter&#x2019;s dynamics in grid-forming mode can be represented, and this is another contribution of this paper. Finally, various studies are presented to prove that the adopted approximated MMC model is accurate enough for small-signal stability analyses.

Reduced-Order Analytical Models of Grid-Connected Solar Photovoltaic Systems for Low-Frequency Oscillation Analysis

Low-frequency oscillations have been observed in a real-world solar photovoltaic (PV) farm. The goal of this research is to build a simplified analytical model in the synchronous frame for large-signal simulation and small-signal analysis. The latter, e.g., eigenvalue analysis and participation factor analysis, can reveal influencing factors of the oscillations. Two simplified analytical models are proposed, ignoring PV dc-side dynamics (e.g., dc/dc boost converter control and maximum power point tracking), while preserving the grid-side converter (GSC) controls and electromagnetic dynamics of the grid interconnection. The first model assumes the input power from PV's dc side known and constant. The second model assumes that PV's dc side is represented by a constant dc voltage behind an impedance. The simplified models are compared with an electromagnetic transient (EMT) testbed with full details on time-domain simulation results, admittance frequency-domain responses, and eigenvalue-based stability analysis results. The second simplified model is found as capable of accurately predicting oscillation stability.

State Space Modeling and Stability Analysis of a VSC-HVDC System for Exchange of Energy

Nowadays, relevant actors are searching for solutions to produce energy with a low impact on the environment. Indeed, transmitting power via large distances with maintaining low losses is one of the main challenges. To improve electricity communication between countries and offshore wind, a new interconnections line must be built. Therefore, Voltage Source Converter High Voltage Direct Current (VSC-HVDC) transmission is incoming as the exceptive technology in order to address the challenges related to the integration of future offshore wind power plants. In spite of its many advantages, VSC-based HVDC transmission systems can experience unexpected instability and interaction phenomena: Small disturbances that occur continually in VSC-HVDC transmission systems due to the complex VSC based interconnections and an important number of components with non-linear nature that may cause failure. Thus, before installing the HVDC system, there is a significant need for studying a hybrid AC-DC system to guarantee the reliable and stable operation. This paper deals with the stability of a VSC-HVDC system by the use of a small signal stability method; such procedure enables to study the stability of a linearized VSC-HVDC system through state-space modeling and eigenvalue-based stability analysis.

A Gray-Box Hierarchical Oscillatory Instability Source Identification Method of Multiple-Inverter-Fed Power Systems

This article presents a gray-box hierarchical instability source identification method of multiple-inverter-fed power systems, which enables stability analysis at system, component, and parameter levels sequentially. Impedance frequency responses of all components are first obtained using frequency scanning method. System impedance network model is then established by connecting these individual components based on system topology, which is further lumped into a loop impedance model (LIM). The vector fitting (VF) algorithm is then used to generate system state–space model from the impedance frequency responses of the LIM for eigenvalue-based stability analysis. If the system is assessed to be unstable, problematic components are further identified by performing impedance-based stability criterion at terminals of all components, where the numbers of right-half-plane poles are obtained by the VF algorithm. Finally, circuit and controller parameters of the identified problematic components are further identified using the VF algorithm, which are retuned to improve system stability. The proposed hierarchical instability source identification method is implemented in a multiple-paralleled grid-connected inverter system. Simulation results obtained in MATLAB/Simulink platform and real-time simulation verification results obtained in OPAL-RT platform are given to validate the correctness of the theoretical analysis results.

The Application of Vector Fitting to Eigenvalue-Based Harmonic Stability Analysis

Participation factor analysis is an interesting feature of the eigenvalue-based stability analysis in a power system, which enables the developers to identify the problematic elements in a multivendor project like in an offshore wind power plant. However, this method needs a full state-space (SS) model of the elements that is not always possible to have in a competitive world due to confidentiality. In this paper, by using an identification method, the SS models for power converters are extracted from the provided data by the suppliers. Some uncertainties in the identification process are also discussed and solutions are proposed, and in the end the results are verified by time domain simulations for linear and nonlinear cases with different complexities, no matter which domain (phase or dq) is used.

Nonlinear vibration of permanent magnet synchronous motors in electric vehicles influenced by static angle eccentricity

An additional transverse electromagnetic moment caused by the inclination of the rotor of the permanent magnet synchronous motors in electric vehicles is modeled based on Maxwell stress tensor method. The model of this moment is suitable for the case of pole-pair number larger than 3 and is independent of time; its nonlinearity results in multiple equilibrium points of the conservative generalized force: an isolated equilibrium point and a continuum of equilibrium points. The continuum is symmetric with respect to the isolated equilibrium point; a pitchfork bifurcation occurs as the system stiffness of the rotor in the non-inclined state passes through zero. When static angle eccentricity is introduced, the symmetry is lost and the continuum degrades into two unstable equilibrium points. This parameter leads to a generic bifurcation with defect. However, considering it as a bifurcation parameter, a pair of saddle-node bifurcations appears. Eigenvalue-based stability analysis and center manifold theorem are employed to determine the stabilities of the multiple equilibrium points. The frequency characteristics of the forced response caused by the static angle eccentricity are developed by harmonic balance method, and the stability of the steady-state solution is studied with Routh–Hurwitz criterion in a rotating system. The frequency response curve generally passes through all the equilibrium points. For a disk-shaped rotor, the frequency characteristics exhibit hardening and the solution has the same stability as the equilibrium point in the same branch. For a cylindrical rotor, the characteristics show softening and only the solution running near the equilibrium point has the same stability as the equilibrium point in the same branch. Furthermore, there is a frequency band in which the forced response is globally unstable. However, for comparatively large damping, large inertia ratio, small stiffness ratio and small static angle eccentricity, the frequency characteristics of a cylindrical rotor are similar to those of a disk-shaped rotor. The globally unstable frequency band diminishes and then vanishes. All of the solutions have the same stability in a branch.

Small-Signal Stability Assessment of Power Electronics Based Power Systems: A Discussion of Impedance- and Eigenvalue-Based Methods

This paper investigates the small-signal stability of power electronics-based power systems in frequency domain. A comparison between the impedance-based and the eigenvalue-based stability analysis methods is presented. A relation between the characteristics equation of the eigenvalues and poles and zeros of the minor-loop gain from the impedance-based analysis have been derived analytically. It is shown that both stability analysis methods can effectively determine the stability of the system. In the case of the impedance-based method, a low phase-margin in the Nyquist plot of the minor-loop gain indicates that the system can exhibit harmonic oscillations. A weakness of the impedance method is the limited observability of certain states given its dependence on the definition of local source-load subsystems, which makes it necessary to investigate the stability at different subsystems. To address this limitation, the paper discusses critical locations where the application of the method can reveal the impact of a passive component or a controller gain on the stability. On the other hand, the eigenvalue-based method, being global, can determine the stability of the entire system; however, it cannot unambiguously predict sustained harmonic oscillations in voltage source converter (VSC) based high voltage dc (HVdc) systems caused by pulse-width modulation (PWM) switching. To generalize the observations, the two methods have been applied to dc-dc converters. To illustrate the difference and the relation between the two-methods, the two stability analysis methods are then applied to a two-terminal VSC-based HVdc system as an example of power electronics-based power systems, and the theoretical analysis has been further validated by simulation and experiments.

Harmonic Instability Assessment Using State-Space Modeling and Participation Analysis in Inverter-Fed Power Systems

This paper presents a harmonic instability analysis method using state-space modeling and participation analysis in the inverter-fed ac power systems. A full-order state-space model for the droop-controlled distributed generation (DG) inverter is built first, including the time delay of the digital control system, inner current and voltage control loops, and outer droop-based power control loop. Based on the DG inverter model, an overall state-space model of a two-inverter-fed system is established. The eigenvalue-based stability analysis is then presented to assess the influence of controller parameters on the harmonic instability of the power system. Moreover, the harmonic-frequency oscillation modes are identified, where participation analysis is presented to evaluate the contributions of different states to these modes and to further reveal how the system gives rise to harmonic instability. Based on the participation analysis, a reduced-order model for harmonic instability analysis is also proposed. The experimental results are presented for validating the theoretical analyses.

Nonlinear dynamic behaviors of permanent magnet synchronous motors in electric vehicles caused by unbalanced magnetic pull

Unbalanced magnetic pull (UMP) plays a key role in nonlinear dynamic behaviors of permanent magnet synchronous motors (PMSM) in electric vehicles. Based on Jeffcott rotor model, the stiffness characteristics of the rotor system of the PMSM are analyzed and the nonlinear dynamic behaviors influenced by UMP are investigated. In free vibration study, eigenvalue-based stability analysis for multiple equilibrium points is performed which offers an insight in system stiffness. Amplitude modulation effects are discovered of which the mechanism is explained and the period of modulating signal is carried out by phase analysis and averaging method. The analysis indicates that the effects are caused by the interaction of the initial phases of forward and backward whirling motions. In forced vibration study, considering dynamic eccentricity, frequency characteristics revealing softening type are obtained by harmonic balance method, and the stability of periodic solution is investigated by Routh–Hurwitz criterion. The frequency characteristics analysis indicates that the response amplitude is limited in the range between the amplitudes of the two kinds of equilibrium points. In the vicinity of the continuum of equilibrium points, the system hardly provides resistance to bending, and hence external disturbances easily cause loss of stability. It is useful for the design of the PMSM with high stability and low vibration and acoustic noise.

Stability Analysis and Convergence Control of Iterative Algorithms for Reliability Analysis and Design Optimization

Iterative algorithms are widely applied in reliability analysis and design optimization. Nevertheless, phenomena of failed convergence, such as periodic oscillation, bifurcation, and chaos, are oftentimes observed in iterative procedures of solving some nonlinear problems. In the present paper, the essential causes of numerical instabilities including periodic oscillation and chaos of iterative solutions are revealed by the eigenvalue-based stability analysis of iterative schemes. To understand and control these instabilities, the stability transformation method (STM), which is capable of tackling numerical instabilities of iterative algorithms in reliability analysis and design optimization, is proposed. Finally, several benchmark examples of convergence control of PMA (performance measure approach) for probabilistic analysis and the SORA (sequential optimization and reliability assessment) for reliability-based design optimization (RBDO) are presented. The observations from the benchmark examples indicate that the STM is a promising approach to achieve convergence control for iterative algorithms in reliability analysis and design optimization.

On Distributed Multirate Control of Direct User-to-User Touch in Networked Haptic Systems with Passive Wave-Domain Communications

This paper investigates the stability and performance of distributed multirate control of direct touch in networked haptic systems that provide users with a remote dynamic proxy of their peer and with passive wave domain communications. The paper considers communication networks with fixed delay and with packet update rate smaller than the update rate of the users' local force feedback loops. After developing a multirate state space model of the direct touch haptic system, the paper uses eigenvalue-based stability analysis to determine the maximum contact stiffness that can be applied to users, as well as the maximum coordination gain that can be used to synchronize the user sites. The analysis predicts that both remote dynamic proxies and passive wave-domain communications make the contact stiffness robust to delay, but only passive wave-domain communications mitigate the negative impact of delay on the coordination gain. In other words, the analysis suggests that passive wave-domain communications should be employed to make the distributed multirate control of direct touch in networked haptic systems stable regardless of the fixed communication delay. However, if power-domain communications are employed, remote dynamic proxies should be used to allow rendering of stiffer contact between users. Experiments in which two users probe each other using similar haptic interfaces validate the analytical results.

Stability analysis of an hydropower generator subjected to unbalanced magnetic pull

Eccentricity leading to unbalanced magnetic pull (UMP) in electrical machines is a significant concern in industry. The UMP is known to be composed of two components: A radial component and a tangential one. Models that are used in industry currently tend to include the radial component alone. In this study, a Jeffcott rotor model together with a new UMP model that incorporates both the radial and tangential UMP constituents is studied for an industrial hydropower generator. Characterising the UMP as springs permits the proposed model to inherit UMP stiffness contribution. It is shown firstly that the new UMP model is sensitive to forcing frequency in the rotor movements and secondly, that this sensitivity to forcing frequency increases with decreasing rotor system stiffness. Moreover, quasi-periodic motion in the rotor displacements is observed for some forcing frequencies and system stiffnesses. Eigenvalue-based stability analysis is performed and shows that damping and stiffness of the rotor and of the bearings are important when non-synchronous whirling of the rotor comes into play. Accounting for both UMP components is an important cornerstone in the generation of better rotor designs which can help to curb rotor–stator malfunction and can contribute in the design of long lasting rotors.