Multi-time Scale Systems Research Articles

Stability enhancement for seamless control in networked microgrids with energy storage: Nonlinear spatio-temporal control perspective

The integration of distributed energy resources into modernized networked microgrids, combined with the increasing variability in load dynamics, presents significant stability challenges. This research offers a comprehensive analysis of the stability of cascaded interactions in nonlinear multi-timescale systems, including grid-following and grid-forming inverters. The proposed grid-forming controller, integrated with energy storage systems and a nonlinear Lyapunov function, facilitates seamless control and stabilization of these inverters. This approach addresses challenges related to nonlinear plant and network dynamics, uncertainties in state variables, and unmodeled system dynamics. The closed-loop control system ensures asymptotic spatio-temporal dynamical stability through constrained time state convergence to stable equilibrium points while minimizing control oscillations. Comparative analysis and high-fidelity hardware-in-loop emulations demonstrate the superior performance of the proposed controller over traditional ones in diverse dynamic scenarios. Its ability to rapidly reject disturbances, minimize overshoot, converge efficiently, and enhance power quality confirms its effectiveness and highlights its potential for real-world applications.

Recovery of synchronized oscillations on multiplex networks by tuning dynamical time scales

The heterogeneity among interacting dynamical systems or variations in the pattern of their interactions occur naturally in many real complex systems. Often they lead to partially synchronized states like chimeras or oscillation suppressed states like in-homogeneous or homogeneous steady states. In such cases, it is a challenge to get synchronized oscillations in spite of prevailing heterogeneity. In this study, we present a formalism for controlling multi layer, multi timescale systems and show how synchronized oscillations can be restored by tuning the dynamical time scales between the layers. Specifically, we use the model of a multiplex network, where the first layer of coupled oscillators is multiplexed with an environment layer, that can generate various types of chimera states and suppressed states. We show that by tuning the time scale mismatch between the layers, we can revive the synchronized oscillations. We analyse the nature of the transition of the system to synchronization from various dynamical states and the role of time scale mismatch and strength of inter layer coupling in this scenario. We also consider a three-layer multiplex system, where two system layers interact with the common environment layer. In this case, we observe anti synchronization and in-homogeneous steady states on the system layers and by tuning their time scale difference with the environment layer, they undergo transition to synchronized oscillations.

Formula omitted] control of singularly perturbed systems using deficient hidden semi-Markov model

This paper deals with the H∞ control of a class of stochastic multi-timescale systems, called Markov jump singularly perturbed systems. The hidden semi-Markov model is introduced to handle the situation when system modes are unavailable in semi-Markov systems. Such a model is assumed deficient, that is, it lacks knowledge about the emission probability, transition probability, and probability density function of the sojourn time. It is a more general case compared with works conducted with perfect transition information. Depending on whether a fast or slow sampling rate is used, the resulting discrete-time singularly perturbed system is modeled differently, for both of which the controller design is conducted. Furthermore, criteria expressed in terms of linear matrix inequalities (LMIs) are developed that guarantee the δ-error mean-square stability. An approach to estimate the upper bound on δ-error with incomplete information is provided, meanwhile, the relationship between system performance and the upper of singular perturbation parameter is also presented. Finally, two simulation examples using real-world systems are provided to corroborate the validity as well as the practical merits of the results.

Multi-level data-predictive control for linear multi-timescale processes with stability guarantee

Multi-timescale dynamics are common in chemical processes. These processes are often difficult to model and pose challenges in control system design. In this paper, we propose a data-based control approach for linear multi-timescale systems using a system behavioural framework. A data resampling method coupled with a novel data predictive control (DPC) design with multi-level optimisation horizons is developed to handle different timescales. To deal with the dynamics of different timescales, the optimisation horizons with small to large time intervals are used to predict and optimise control actions from near to distant future. Computational complexity wise, the multi-level structure allows horizon length to expand exponentially with optimisation steps. A trajectory-based dissipativity condition is also developed to ensure stability of the proposed DPC, while achieving disturbance rejection and tracking control. An example of controlling a multi-timescale reactive distillation column is presented to illustrate the proposed approach.

Short-Duration Transient Temperature Distribution Prediction Model along Chip Vertical Path Applicable to Multi-Timescale Simulation

Based on extremely uneven temperature distribution along the insulated gate bipolar transistor (IGBT) chip vertical path during switching transients, a short-duration transient microsecond-scale prediction model applicable to multi-timescale simulation is presented in this paper. Traditional thermal models often take the chip active area as a uniform heat source to obtain a victual junction temperature (Tvj). In this paper, a discrete distributed heat source model combined with a thermal network model based on the sublayer division strategy is proposed to achieve an accurate temperature distribution description along the chip vertical path. Taking a 1700 V/3600 A IGBT module as an example, the proposed model can evaluate the short-duration transient temperature distribution along the chip vertical path and the error is less than 2 °C compared with the finite element model. Meanwhile, the model is applied to a single short-duration transient timescale and multi-timescale systems separately, and the simulation speed is increased by more than 80 times and 297 times, which verifies its validity and accuracy.

Bursting Oscillations and Experimental Verification of a Rucklidge System

Bursting oscillations are ubiquitous in multi-time scale systems and have attracted widespread attention in recent years. However, research on experimental demonstration of the bursting oscillations induced by delayed bifurcation is very rarely reported. In this paper, a parametrically driven Rucklidge system is introduced and a distinct delayed behavior is observed when the time-varying parameter passes through the pitchfork bifurcation point. Different bursting patterns induced by such a delayed behavior are numerically investigated under different excitation amplitudes based on the fast–slow analysis method. Furthermore, in order to reproduce the bursting electronic signals and explore the underlying formation mechanisms experimentally, a real physical circuit of the parametrically driven Rucklidge system is developed by using off-the-shelf electronic devices. The real-time measurement results such as time series, phase portraits and transformed phase portraits are in good qualitative agreement with those obtained from the numerical computations. The experimental evidence to verify bursting oscillations induced by delayed pitchfork bifurcation is thus provided in this study.

Delayed Transcritical Bifurcation Induced Mixed Bursting in a Modified SM System with Asymmetrically Distributed Equilibria

Delayed bifurcation behavior is ubiquitous in multi-time scale systems and has a great important effect on bursting oscillations. In this paper, a parametrically driven Shimizu-Morioka system is proposed and distinct delay behavior is observed when the slow-varying parameter passes through the transcritical bifurcation point periodically. Periodic and chaotic bursting oscillations induced by such delay behavior are investigated by using the fast-slow analysis method with different excitation amplitudes. More interesting, the fast subsystem has a zero equilibrium point branch and two asymmetrically distributed equilibrium point branches with respect to the slow-varying parameter. Thus, there exist two possible routes for the trajectory to choose when the delayed transcritical bifurcation takes place. This leads to that the bursting patterns corresponding to different excitation periods may be different. As a result, two possible mixed bursting patterns, named as “delayed transcritical/transcritical” bursting of point-point type mixed with “delayed transcritical/supHopf/fold” bursting of point-cycle type and “delayed transcritical/subHopf/subHopf/fold” bursting of point-cycle type mixed with “delayed transcritical/supHopf/fold” bursting of point-cycle, are revealed. Furthermore, the effect of the excitation frequency on the delay behavior is also considered. We find the delay behavior may terminate at different parameter areas when the excitation amplitude is fixed and the excitation frequency varies and thus leads to different bursting patterns. Numerical simulations are provided to verify the validity of the study.

Model Order Reduction with True Dominant Poles Preservation via Particles Swarm Optimization

A new computational technique for the reduction of multi-time scale systems is proposed in this paper. The reduction process is performed based on the dominant poles preservation in the reduced-order model. The true dominant poles are selected based on the highest contribution in redefined time moments and lowest contribution in redefined Markov parameters. Motivated by the singular perturbation approximation, obtaining the reduced-order model will be achieved by using the artificial intelligent method named particles swarm optimization. The potential of the proposed technique is observed when comparing its results with other recently published methods.

The Lyapunov-based stability analysis of reduced order micro-grid via uncertain LMI condition

This paper proposed a new three-step method for studying dynamic stability of an islanded micro-grid (MG). In conventional methods, state-space model linearization around an operating point and modal analysis are utilized. Due to lower inertia and operation of MG in islanding mode, MG is more sensitive to changes of parameters and contingencies. Therefore, small signal stability (sss) analysis has a very limited validity range and large signal studies should be also performed to get an appropriate perception of the system stability. In this paper, beside sss study to determine network stability status around a dominant operating point, an algorithm based on multi-time scale systems theory is suggested to detect slow and fast modes which can be used for system decomposition into fast and slow subsystems. Fast modes have large negative real values and/or damping ratios, which can be eliminated in large signal stability analysis with an appropriate approximation. Therefore, singular perturbation theory is used to reduce the model order and extract the network slow subsystem. Then, the Lyapunov function can be applied to reduced model. In addition, continuous load and the renewable distributed generators power variations in the system cause to wide range of changes in the operating point and state space model of the system. Therefore, the copula distribution is proposed to model the uncertainties. Then, the uncertain LMI condition based on scenarios extracted from the copula is used to determination of the Lyapunov function and the attraction domain. Finally, the attraction domain sensitivity to MG parameters change is studied to achieve sufficient insight about the system stability margins.

Firefly artificial intelligence technique for model order reduction with substructure preservation

Model order reduction (MOR) is a process of finding a lower order model for the original high order system with reasonable accuracy in order to simplify analysis, design, modeling and simulation for large complex systems. It is desirable that the reduced order model preserves the fundamental properties of the original system. This paper presents a new MOR technique of multi-input multi-output systems utilizing the firefly algorithm (FA) as an artificial intelligence technique. The reduction operation is proposed to maintain the exact dominant dynamics in the reduced order model with the advantage of substructure preservation. This is mainly possible for systems that are characterized as multi-time scale systems. Obtaining the reduced order model is achieved by minimizing the fitness function that is related to the error between the full and reduced order models’ responses. The new approach is compared with recently published work on firefly optimization for MOR, in addition to three other artificial intelligence techniques; namely, invasive weed optimization, particle swarm optimization and genetic algorithm. As a result, simulations show the potential of the FA for the process of MOR.

A cycle-based formulation for the simulation of multi time-scale systems — Application to the modeling of the storage system of a fully electric ferry

This paper addresses the simulation of complex systems which consider phenomena with different time scales. Such problems are encountered on studies of electrical systems which try to take into account the simulation of the power converter and its control laws, over representative operating cycles of several hours. Moreover, when storage elements are integrated into a power chain, their aging may need the designer to consider larger time scales, which can then exceed a few years. It is the reason why most studies separate the time scales between the slow dynamics for the energetic, thermal and aging phenomena, and the fast ones to study the power converter and its control laws. This paper presents an original cycle-based and multi-rate method for the simulation of power systems with a wide range of time scales and with high mutual dependency between the fast and slow state variables. This method is applied to the supercapacitor energy storage system of a full-electric ferry. The proposed simulation results take into account at the same time the switching of the power converter and the aging of the supercapacitor, with a reduction of the computational effort greater than 105. In other words, while a full calculation of the problem takes 10 centuries on a personal computer, the proposed method permits to have the same result in only 15 days.

Multi-timescale systems and fast-slow analysis

Mathematical models of biological systems often have components that vary on different timescales. This multi-timescale character can lead to problems when doing computer simulations, which can require a great deal of computer time so that the components that change on the fastest time scale can be resolved. Mathematical analysis of these multi-timescale systems can be greatly simplified by partitioning them into subsystems that evolve on different time scales. The subsystems are then analyzed semi-independently, using a technique called fast-slow analysis. In this review we describe the fast-slow analysis technique and apply it to relaxation oscillations, neuronal bursting oscillations, canard oscillations, and mixed-mode oscillations. Although these examples all involve neural systems, the technique can and has been applied to other biological, chemical, and physical systems. It is a powerful analysis method that will become even more useful in the future as new experimental techniques push forward the complexity of biological models.

Two-Stage Feedback Control Design for a Class of Linear Discrete-Time Systems With Slow and Fast Modes

In this paper, we first review the new algorithm for the two-stage feedback controller design of linear discrete-time systems, and then provide conditions for its applicability. The design algorithm is specialized and simplified for a class of linear systems with slow and fast modes (multitime scale systems or singularly perturbed systems). The proposed design significantly reduces computational full-state feedback design requirements and provides independent and accurate feedback controller design techniques in slow and fast time scales. We present also conditions needed for applicability of the proposed two-stage design in two time scales. The power of the two-stage design lies in the fact that different types of controllers can be designed for different subsystems using the corresponding feedback gains obtained by performing calculations only with the subsystem (reduced-order) matrices.

Soft Computing Techniques for Reduced Order Modelling: Review and Application

AbstractAs the mathematical procedure of system modelling often leads to a comprehensive description, which causes significant difficulty in both analysis and control synthesis, it is necessary to find lower order models, which maintain the dominant characteristics of the original system. In this paper, different soft computing (named as artificial intelligence (AI)) techniques are presented, applied, and analysed for model order reduction (MOR) of multi time scale systems with the objective of substructure preservation. In addition to that, we investigate the firefly optimization technique for MOR with substructure preservation. The analysis is concerned with the optimization approach and quality of method performance.

Vibration mitigation of a bridge cable using a nonlinear energy sink: design and experiment

This work deals with the design and experiment of a cubic nonlinear energy sink (NES) for horizontal vibration mitigation of a bridge cable. Modal analysis of horizontal linear modes of the cable is experimentally performed using accelerometers and displacement sensors. A theoretical simplified 2-dof model of the coupled cable-NES system is used to analytically design the NES by mean of multi-time scale systems behaviours and detection its invariant manifold, equilibrium and singular points which stand for periodic and strongly modulated regimes, respectively. Numerical integration is used to confirm the efficiency of the designed NES for the system under step release excitation. Then, the prototype system is built using geometrical cubic nonlinearity as the potential of the NES. Efficiency of the prototype system for mitigation of horizontal vibrations of the cable under for step release and forced excitations is experimentally demonstrated.

A nonlinear sliding mode observer for the estimation of temperature distribution in a planar solid oxide fuel cell

Compliance to certain temperature constraints is essential for the long life and stable operation of solid oxide fuel cells (SOFCs). However, it is difficult and costly to measure the temperature inside a high temperature, well-sealed cell directly. A nonlinear sliding mode observer has been designed in this paper for estimating the temperature distribution in a hydrogen fed SOFC stack. The observer design is based on a finite-volume parameter SOFC model, which is simplified by quasi-static mass balance assumption. In the observer design, a decoupling pole placement method that applies to multi-timescale systems is proposed to obtain a suitable observer gain. And a quasi-sliding mode control method is adopted to eliminate the chatter caused by the discontinuous control actions of sliding mode control. Theoretical analysis and simulation results are presented to prove the convergence and good performance of the designed observer for estimating the temperature distribution in a SOFC stack during steady-state and transient operation.

A Simplified Two-Stage Design of Linear Discrete-Time Feedback Controllers

In this paper, we have shown how to simplify an algorithm for the two-stage design of linear feedback controllers by reducing computational requirements. The algorithm is further simplified for linear discrete-time systems with slow and fast modes (multitime scale systems or singularly perturbed systems), providing independent and accurate designs in slow and fast time scales. The simplified design procedure and its very high accuracy are demonstrated on the eigenvalue assignment problem of a steam power system.

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, multiple time scales, or chaotic dynamics. In particular, we prove that the differential equations for the coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of truncation order. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincare section of the system and that, as the scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

A Homogenization Approach to Multiscale Filtering

We present a homogenized nonlinear filter for multi-timescale systems, which allows the reduction of the dimension of filtering equation. We prove that the actual nonlinear filter converges to our homogenized filter. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and probabilistically representing the correction terms with the help of backward doubly-stochastic differential equations. This homogenized filter provides a rigorous mathematical basis for the development of reduced-dimension nonlinear filters for multiscale systems. A filtering scheme, based on the homogenized filtering equation and the technique of importance sampling, is applied to a chaotic multiscale system in Lingala et al. [1].

Robust decentralized eigenvalues clustering for multi-time-scale systems with perturbations

The formulation of the robust decentralized eigenvalues clustering with the design technique of the decentralized robust state feedback controller for multi-time-scale systems is facilitated. The multi-time-scale system, consisting of a slow subsystem and some fast subsystems with perturbations, is considered. From the eigenvalues locations of the multi-time-scale system, it is known that the locations of the fast subsystems are far from the slow subsystem ones. Furthermore, presented here are some sufficient conditions of robust eigenvalue clustering in specified regions for the uncertain slow subsystem and the uncertain fast subsystems with the parameter perturbations. An algorithm for the design of a decentralized robust state feedback controller for multitime-scale systems is then presented. It not only ensures that the eigenvalues of these subsystems are located in the desired regions, but also makes sure that all the eigenvalues of the whole system are in our desired regions. An illustrative example is adopted, and the result is satisfactory.