Fast Subsystem Research Articles

Reduced-order modeling of DFIG-based wind turbine connected into weak AC grid based on electromechanical time scale

Nowadays, the stability issue of the doubly fed induction generator (DFIG)-based wind turbines tied into weak AC power grid has attracted more and more attention. To analyze the stability of such cascade system, it is a common way to establish the small-signal model of the whole system. Although there have been various modeling methods, few of them can be easily utilized in practice due to the high order of the presented models. To address this issue, this paper simplifies the fast time scale subsystems and proposes a reduced-order state space model of the DFIG system based on electromechanical time scale, considering the rotor current controller (RCC), the phase-locked loop (PLL), the remained value of the grid voltage, and the impedance of the weak AC grid as well. Based on the simplified model, the stability of the wind turbine during weak AC grid voltage dips can be easily analyzed. The correctness of the integrated model and its convenience for stability analysis have been proved by eigenvalue matching method and computer simulations.

An asynchronous MATE-multirate method for the modeling of electric power systems

The multiarea Thevenin equivalent (MATE) solution framework is used in this work to combine several subsystems, which are solved using different multiple and non-multiple time-step sizes. In the proposed asynchronous MATE-multirate (A-MATE-multirate) solution, each subsystem is solved with the integration step size that is most adequate for its own internal time constants. This paper relaxes the requirement of previous multirate techniques, in which the solution time-steps of different systems must be multiples or submultiples of each other. To this end, this paper introduces the concept of asynchronous updating, in which each subsystem is updated according to its own internal clock. This updating process is performed in a closed-form solution, with external subsystems represented by their interpolated Thevenin equivalents. Interpolation with future values without using extrapolation is possible in the electromagnetic transients program solution because the history source for future time-steps is calculated at previous time-steps. The exchange of information between subsystems is conditioned using filters to avoid numerical problems during the decimation of the fast subsystems and the interpolation of the slow subsystems. The effectiveness of the proposed methodology is verified with examples using multiple and non-multiple time-step sizes.

On occurrence of bursting oscillations in a dynamical system with a double Hopf bifurcation

The slow-fast effect in a vector field with a codimension-two double Hopf bifurcation at the origin is investigated in the paper. To explore the typical evolution of the dynamics, the universal unfolding of the normal form of the vector field is taken into consideration. When the parametric excitation is introduced, the frequency of which is far less than the two natural frequencies, slow-fast behaviors may appear. Regarding the whole parametric excitation term as a slow-varying parameter, the equilibrium branches and the bifurcations of the generalized autonomous fast subsystem can be derived. With the variation of the exciting amplitude, different types of bursting oscillations caused by the coupling of two scales in frequency domain may appear, the mechanism of which are obtained by employing the overlap of the transformed phase portraits and the equilibrium branches as well as the bifurcations of the fast subsystem. It is found that, for relatively small exciting amplitude, no bifurcation of the fast subsystem occurs, and the system behaves in quasi-periodic oscillations. With the increase of the exciting amplitude, different types of bifurcations may take place, leading to the single-mode bursting oscillations, which may evolve to mixed-mode bursting oscillations. Symmetric breaking bifurcations result in the transitions between a symmetric attractor and two co-existed asymmetric attractors in the system. Furthermore, for the mixed-mode bursting oscillations, the trajectory may alternate between the single-mode oscillations and the mixed-mode oscillations via bifurcations of the equilibrium branches, causing the synchronization and non-synchronization between different state variables.

Two bursting patterns induced by system solutions approaching infinity in a modified Rayleigh–Duffing oscillator

In this paper, the mechanism of system solutions approaching infinity is explored based on a modified Rayleigh–Duffing oscillator with two slow-varying periodic excitations. System solutions approaching infinity is a new novel route to bursting oscillation, and are not reported yet. The system can be separated into a fast subsystem and a slow subsystem according to the slow–fast analysis method. We find that there is a critical value for the fast subsystem, which limits the original region of the stable equilibrium point and the stable limit cycle, the right of which is the divergent region. When the control parameter slowly varies closely to the critical value $$\delta _{\mathrm{CR}} $$ , both the stable equilibrium point and the stable limit cycle quickly leave the original region and approach positive infinity. The mechanism of two different bursting forms called bursting oscillation of point/point and bursting oscillation of cycle/cycle induced by system solutions approaching infinity are explored. This paper provides a new possible route to bursting oscillation unrelated to bifurcations and deepens the comprehension of bursting dynamics behaviours. Lastly, the accuracy of our study is verified by overlapping the transformed phase portraits onto the bifurcation diagrams.

Control of a flexible inverse pendulum based on the singular perturbation method

In this article, a partial differential equation model for a flexible inverted pendulum system is derived by using the Hamilton principle. Specifically, problems of stabilization and the optimization of such a system are considered. In addition, the singular perturbation method has been used to divide the partial differential equation model for a fast and a slow subsystem. For a fast subsystem stabilization, the control algorithm proposed a boundary force applied at the free end of the beam which proved that the closed-loop subsystem is appropriate and exponentially stable. To stabilize the slow subsystem, a sliding mode control method was used to design the controller, while the method of linear matrix inequality was used in designing the sliding surface. In conclusion, it was shown that the slow subsystem is exponentially stable.

Bursting during intermittency route to thermoacoustic instability: Effects of slow-fast dynamics.

Intermittency observed prior to thermoacoustic instability is characterized by the occurrence of bursts of high-amplitude periodic oscillations (active state) amidst epochs of low-amplitude aperiodic fluctuations (rest state). Several model-based studies conjectured that bursting arises due to the underlying turbulence in the system. However, such intermittent bursts occur even in laminar and low-turbulence combustors, which cannot be explained by models based on turbulence. We assert that bursting in such combustors may arise due to the existence of subsystems with varying timescales of oscillations, thus forming slow-fast systems. Experiments were performed on a horizontal Rijke tube and the effect of slow-fast oscillations was studied by externally introducing low-frequency sinusoidal modulations in the control parameter. The induced bursts display an abrupt transition between the rest and the active states. The growth and decay patterns of such bursts show asymmetry due to delayed bifurcation caused by slow oscillations of the control parameter about the Hopf bifurcation point. Further, we develop a phenomenological model for the interaction between different subsystems of a thermoacoustic system by either coupling the slow and fast subsystems or by introducing noise in the absence of slow oscillations of the control parameter. We show that interaction between subsystems with different timescales leads to regular amplitude modulated bursting, while the presence of noise induces irregular amplitude modulations in the bursts. Thus, we speculate that bursting in laminar and low-turbulence systems occurs predominantly due to the interdependence between slow and fast oscillations, while bursting in high-turbulence systems is predominantly influenced by the underlying turbulence.

Θ-SEIHRD mathematical model of Covid19-stability analysis using fast-slow decomposition.

In general, a mathematical model that contains many linear/nonlinear differential equations, describing a phenomenon, does not have an explicit hierarchy of system variables. That is, the identification of the fast variables and the slow variables of the system is not explicitly clear. The decomposition of a system into fast and slow subsystems is usually based on intuitive ideas and knowledge of the mathematical model being investigated. In this study, we apply the singular perturbed vector field (SPVF) method to the COVID-19 mathematical model of to expose the hierarchy of the model. This decomposition enables us to rewrite the model in new coordinates in the form of fast and slow subsystems and, hence, to investigate only the fast subsystem with different asymptotic methods. In addition, this decomposition enables us to investigate the stability analysis of the model, which is important in case of COVID-19. We found the stable equilibrium points of the mathematical model and compared the results of the model with those reported by the Chinese authorities and found a fit of approximately 96 percent.

Projection-based adiabatic elimination of bipartite open quantum systems

Adiabatic elimination methods allow the reduction of the space dimension needed to describe systems whose dynamics exhibit separation of timescale. For open quantum systems, it consists in eliminating the fast part assuming it has almost instantaneously reached its steady state and obtaining an approximation of the evolution of the slow part. These methods can be applied to eliminate a linear subspace within the system Hilbert space or, alternatively, to eliminate a fast subsystem in a bipartite quantum system. In this work, we extend an adiabatic elimination method used for removing fast degrees of freedom within a system [Phys. Rev. A 101, 042102 (2020)] to eliminate a subsystem from an open bipartite quantum system. As an illustration, we apply our technique to a dispersively coupled two-qubit system and in the case of the open Rabi model.

Bounded decoupling control for flexible‐joint robot manipulators with state estimation

This study proposes a bounded controller for flexible-joint robot manipulators that only requires state measurements of joint motors. Firstly, the singular perturbation theory is used to decouple the dynamics model of the flexible-joint robot manipulator into slow and fast subsystems. Then, a saturated adaptive scheme combining backstepping control and radial basis function neural network approximator is applied to make the controller bounded and simplify the control law for the slow subsystem, and a new filtering operator is used in the fast subsystem to enhance the overall performance. Furthermore, the extended Kalman filter is involved to eliminate the states measurements of links in the closed-loop tracking control. Finally, the stability of the whole system is analysed and the effectiveness of the proposed approach is verified.

Multi-Rate Real-Time Simulation Method Based on the Norton Equivalent

For the problem of poor accuracy of the existing multi-rate simulation methods, this paper proposes a multi-rate real-time simulation method based on the Norton equivalent, compared with multi-rate simulation method based on the ideal source equivalent. After the Norton equivalence of the fast subsystem and the slow subsystem are established, they are solved simultaneously at the junction nodes. In order to reduce the amount of the simulation calculation, the Norton equivalent circuit is obtained by incremental calculation. The data interaction between the fast subsystem and the slow subsystem is realized by extrapolation method. For ensuring the real-time performance of the simulation, the method of the slow subsystem calculates ahead of the fast subsystem is given for the slow subsystem with a large amount of calculation. Finally, the AC/DC hybrid power system was simulated on the real-time simulation platform (FPGA-based Real-Time Digital Solver, FRTDS), and the simulation results were compared with the single-rate simulation, which verified the correctness and accuracy of the proposed method.

Paradoxical reduction and the bifurcations of neuronal bursting activity modulated by positive self-feedback

Paradoxical enhancement rather than reduction in firing activity induced by inhibitory effect is very important for both nonlinear dynamics and neuroscience. In the present paper, to build a more comprehensive viewpoint related to excitatory effect, a paradoxical phenomenon modulated by the excitatory self-feedback of autapse is extended to bursting activity in a modified Morris–Lecar model. With increasing the strength of excitatory autapse, the bursting patterns exhibit inverse period-adding bifurcations to spiking pattern via chaos, and the burst duration, the spike number per burst, and the firing frequency of bursting decrease, which is different from the traditional view that firing frequency should be enhanced. Furthermore, the paradoxical reduction in bursting activity is well explained with the nonlinear dynamics of the fast subsystem. The burst begins from neighborhood of a fold bifurcation of equilibrium point and terminates at a saddle homoclinic (SH) bifurcation. With increasing conductance of autapse, the fold bifurcation point remains unchanged, which is due to too small autaptic current near the fold point to influence the dynamics, and the SH point shifts left, which follows from the enhanced potassium current at the minimal value of the stable limit cycle. Therefore, the range between the fold and SH point becomes narrower to shorten the burst duration to reduce spike number per burst and firing frequency. The novel example of paradoxical reduction modulated by positive self-feedback and the associated bifurcation mechanism enrich the phenomena of nonlinear science and present the potential functions of excitatory autapse in the brain neurons with bursting behavior.

Three-Phase Shunt Active Filter Control in Sliding Modes

A simple control strategy of the three-phase shunt active power filter that is connected to a non-linear load network is proposed. The filter consists of a three-phase voltage source inverter based on fully controlled switches, capacitor storage, and a first-order three-phase RL filter. The decomposition of the object of study according to the rates of motion of the dynamic system was performed. The fast subsystem is a circuit of compensation current control, the slow subsystem performs stabilization of the capacitor voltage. The basis of this approach is the separation of the vector of state variables into groups of components and their consistent use as control actions. To stabilize the DC voltage on the storage capacitor, a double twisting algorithm, which is based on the forced introduction of a one-dimensional second order sliding mode is used. The proposed strategy provides asymptotic stability in the error metric of the second-order sliding mode . The output signal of the voltage stabilization subsystem is a component of the reference signal for the current tracking circuit. A low pass filter to separate the subsystems is used. Compensation current is forming by the forced entering the first-order sliding mode. To reduce the number of current sensors in the system, an indirect control strategy is used. To generate the compensation current, a two-dimensional sliding manifold, which is a linear combination of components of two-dimensional vectors - the current error of the RL filter and some variable is used. The form of this variable is determined on the assumption that the first-order sliding mode in the current loop already exists and is ideal. Each component of this sliding surface is a linear combination of components of two-dimensional vectors. The first vector is the current error of the RL filter, the coordinates of the second are locally equivalent to the derivative of fundamental harmonic of this error. A simulation model is built and the simulation results are analyzed. The proposed strategy is compared with the traditional PI-regulation on the criteria of the duration of the transient process and the coefficient of harmonic distortion in the current consumed from the grid.

Analysis of a breast cancer mathematical model by a new method to find an optimal protocol for [formula omitted]-positive cancer

Treatment of breast cancer (positive for HER2, i.e., ERBB2) is described by a mathematical model involving non-linear ordinary differential equations with a hidden hierarchy. To reveal the hierarchy of dynamical variables of the system being considered, we applied the singular perturbed vector field (SPVF) method, where a system of equations can be decomposed to fast and slow sub-systems with explicit small parameters. This new form of the model, which is called a singular perturbed system, enables us to apply a semi-analytical method called the method of directly defining inverse mapping (MDDiM), which is based on the homotopy analysis asymptotic method. We introduced the treatment protocol in explicit form, through an analytical function that describes the exact dose and intervals between treatments in a cyclical manner. In addition, a new algorithm for the optimal dosage that causes tumour shrinkage is presented in this study. Furthermore, we took the concept of protocol optimisation a step further and derived a differential equation that represents vaccination depending on tumour size and yields an optimal protocol of different doses at every time point. We introduced the treatment protocol in explicit form, through an analytical function that describes the exact dose and intervals between treatments in a cyclical manner. In addition, a new algorithm for finding the optimal dosage that causes tumour shrinkage is presented in this study. Additionally, we took the concept of protocol optimisation a step further and derived a differential equation that represents vaccination depending on tumour size and yields an optimal protocol of different doses at every time point.

Relaxation and mixed mode oscillations in a shape memory alloy oscillator driven by parametric and external excitations

Abstract This paper performs analytical investigations on symmetric jump phenomena reflecting multi-timescale dynamics in a nonlinear shape memory alloy oscillator with parametric and external cosinoidal excitations by means of geometrical singular perturbation theory (GSPT). The conditions concerning the existence of homoclinic and heteroclinic chaos subjected to periodic perturbations are studied in mathematical models by applying Melnikov method. Then we treat the excitation item as the slow variable acting on the bifurcation structure of the fast subsystem and present explicit algebraic expressions of the critical manifold of this subsystem for the investigation of fast-slow motions in the whole system. The study reveals the dynamical features of the relaxation oscillations formed by the alternative stable states on the different branches. In addition, the range of parametric excitation amplitude and stiffness values of the system plays an important role in the singularities and shaping the oscillation patterns. We determine numerically how the parameter values affect the manifold of the fast subsystem during the active fast transitions. Using the fast-slow analysis, one can provide the correct analytical predictions of the region of parameter space where the periodic orbits that alternate between epochs of fast and slow motions occur. Numerical simulations are also carried out to illustrate the validity of our study.

Quality assurance test of the STS-XYTERv2 ASIC for the silicon tracking system of the CBM experiment

The Compressed Baryonic Matter (CBM) Experiment will be one of the major scientific endeavors of the future Facility of Antiproton and Ion Research (FAIR) in Darmstadt. Nuclear collisions at an interaction rate of up to 10 MHz will provide the data for numerous observables, including rare probes with necessary statistical significance. Key to achieve this goal is a fast and radiation hard detector subsystem for particle tracking. In the CBM experiment, the Silicon Tracking System (STS) is the main system for particle tracking. Signal read-out from the 2x1024 channel sensors is based on the self-triggering STS-XYTERv2 ASIC specifically developed for this application. Towards final STS-operation, a down-scaled mini-CBM (mCBM) setup was taken into operation at the SIS18 accelerator at GSI. In this paper, the test measurements of the STS-XYTERv2 asıcs and operational tests performed for quality assurance during the assembly of the mSTS detector modules will be presented and discussed in detail. These measurement procedures will constitute fundamental processes for quality assurance (QA) in the assembly of the roughly 900 STS detector modules.

Multiple coherence resonances evoked from bursting and the underlying bifurcation mechanism

Coherence resonance (CR), which is typical nonlinear phenomenon that noise plays construct roles, has been widely investigated in single neurons at resting state instead of bursting behavior when noise free. In the present paper, multiple CRs evoked from bursting instead of resting state and the underlying bifurcation mechanism are studied in a theoretical neuron model composed of fast subsystem and slow variable, which is very important for the bursting neurons receiving strong synaptic noise in the central nervous system. With increasing noise intensity within a range, the bursting period decreases to induce the burst number increases firstly and then becomes irregular, which results in the increase firstly and then decrease in the corresponding peak of the power spectrum of membrane potentials, i.e., CR is evoked from bursting instead of resting state. With further increasing noise intensity, two other peaks exhibit CR phenomenon, one at middle frequency and middle noise intensity and the other at large frequency and strong noise intensity, which shows that multiple CRs are evoked from bursting. Furthermore, the bifurcation mechanism of the latter two resonant peaks is acquired with fast–slow variable dissection method. With increasing the value of slow variable, the fast subsystem exhibits a fold bifurcation of limit cycle and a subcritical Hopf bifurcation to a stable focus, and the imaginary part of the characteristic value of the focus increases, which implies that spikes evoked from the focus exhibit increasing frequency. The trajectory of the deterministic bursting in phase space locates between the two bifurcations and alternates between the coexisting limit cycle and focus of the fast subsystem. With increasing noise intensity, the value of slow variable of the stochastic bursting increases and moves to a range of the focus of the fast subsystem, and the spikes of the stochastic bursting are evoked from the focus by noise. Therefore, the stronger the noise intensity, the larger the value of slow variable, and the higher frequency the spikes, due to the changes of characteristic values, which is the cause that the peaks with middle and high frequency appear at middle and strong noise intensity, respectively. The results of the present paper extend the concept of CR to multiple CRs evoked from bursting, provide the bifurcation mechanism underlying the multiple CRs, and imply multiple chances to utilize noise to enhance information of bursting neurons.

Vision‐based tip position tracking control of two‐link flexible manipulator

Owing to the non-collocation of actuators and sensors in a flexible-link manipulator (FLM) it becomes difficult to achieve accurate tip position tracking. To resolve this issue, a vision sensor is used for direct measurement of the tip position instead of employing the traditional mechanical sensors. Among the different visual servoing (VS) control schemes, image-based VS (IBVS) is more effective. However, there are many challenges in the IBVS scheme such as singularities in the interaction matrix and local minima in trajectories that affect the system performance in real-time applications. In this study, the moment-based new visual feature is selected to address the aforesaid issues that arise in the IBVS scheme. Furthermore, a new two-time scale IBVS controller is developed for addressing the tip-tracking control problem of the two-link flexible manipulator (TLFM). In the proposed control scheme, the dynamics of the FLM is decomposed into two-time scale models, namely a slow subsystem and a fast subsystem. The performance and robustness of the proposed new two-time scale IBVS controller for TLFM are verified by pursuing simulation studies. It is observed from the obtained results that the proposed controller effectively stabilises the oscillatory dynamics and tracks the reference trajectory accurately.

Bursting vibration-based energy harvesting

The main purpose of this article is to demonstrate that bursting oscillations can be exploited to enhance the harvested electrical power. A vibration-based bistable Duffing energy harvester, a tristable energy harvester and an asymmetric bistable energy harvester are examined, and bursting oscillations are observed in the energy-harvesting systems with periodic excitation when an order gap exists between the exciting frequency and the natural frequency. The bifurcation mechanism of the bursting oscillations is presented via the bifurcation diagram and the transformed phase portrait of fast subsystem, which reveals that fold bifurcations occur at the transition between the quiescent states and the repetitive spiking states. Then, we investigate the influence of resistive load on the output power, and the optimal resistance is employed to determine the maximum of the power. Furthermore, compared with the method of traditional resonance energy harvesting, results clearly illustrate an improved output power.

Mixed mode oscillations induced by bi-stability and fractal basins in the FGP plate under slow parametric and resonant external excitations

In this paper, three patterns of mixed mode oscillations (MMOs) in the functionally graded piezoelectric (FGP) plate are analyzed. By introducing a slow axial excitation into the two degrees of freedom model, a fast-slow dynamic system with slow parametric and resonant external excitation is established. Using the resonance relations and the multiple scales method, the averaged equations as well as the approximation of stationary responses of the fast sub-system (FSS) are obtained. Thus, predictions of the possible distribution of periodic solutions are given for both two modes. Then numerical bifurcation diagrams of the FSS with respect to the control parameter are computed. It is shown that the FSS can present the coexistence of the stable limit cycle with small amplitude and the multi-period cycle with large amplitude. What is more, the amplitude of the external excitation has little qualitative effects on the bifurcation structure of the FSS. On the other hand, for different amplitude of the external excitation, three patterns of MMOs in the fast-slow system are observed. To explain the corresponding generation mechanisms, fast-slow analysis and basins of attraction (BOAs) of the FSS are combined to depict the transition behaviors of the fast-slow flow. Our results show that, as the amplitude of the external excitation increases, in the vicinity of the small amplitude cycle, the basin of attraction (BOA) of the big amplitude cycle will expand. Meanwhile, since the BOAs possess fractal structures, destinations of these transitions can show three patterns, namely cycle with small amplitude, multi-period cycle with large amplitude, hybrid of small/big cycle. Based on these, generation mechanisms of three MMOs patterns, which can be named as “cycle-chaos-cycle” MMOs, “cycle-chaos-big cycle” MMOs, “cycle-chaos-hybrid of small/big cycle” MMOs, are explained from a numerical view.

Time scale stability analysis of a Hopf bifurcation in a wind–diesel hybrid microgrid

Microgrids and modern bulk power systems usually have multiple time scale dynamics, such as slow and fast dynamics. In this study, the stability of a wind–diesel hybrid microgrid is investigated to show a mechanism of system collapse caused by the interaction between fast dynamics of interface converters and slow electromechanical dynamics. This mechanism leads to a Hopf bifurcation of the fast subsystem due to slow changes in the variables of the slow subsystem and is completely understood by decomposing the stability analysis of the microgrid into the stability assessment of two simpler subsystems: the slow and the fast subsystems. The time-scale method proposed in this study for stability analysis of the microgrid extends the existing proposals in the literature and is able to detect this kind of instability scenario, where the interaction between fast and slow subsystems is the cause of collapse, while the existing approaches, such as quasi-steady-state analysis, fail in detecting instability.