Singular Perturbation Method Research Articles

Research on Reduced-Order Modeling of Droop Controlled Inverter Suitable for Analysis of Transient Characteristics

Under the background of a large number of converters connected to the power grid, the converter’s weak overcurrent capability and poor anti-interference ability bring challenges and hidden dangers to the transient safe and stable operation of the power grid. It is urgent to carry out the transient characteristics analysis of the converter. Numerical simulation calculation is an important means to study transient characteristics. However, numerical simulation is time-consuming and poor in generality, which greatly limits the study of transient characteristics of systems with multiple converters. In this paper, the vertical control is the object, and the inverter reduced-order simplified model suitable for different transient stability problems is studied. First, a full-order model of the droop control inverter is established, and the singular perturbation method is used to complete the extraction of system characteristic roots and the analysis of participating factors. Then, according to the droop control of the inverter’s power angle, voltage, and frequency transient response process, accuracy and rapidity are selected as the goals, and a reduced-order simplified model suitable for different transient stability analysis is obtained. The transient characteristic analysis of the numerical simulation of the converter system provides a reduced order reference.

Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response

In this paper we study the spruce-budworm interaction model with Holling's type II functional response. The existence, number and stability of equilibria are studied. Moreover, we prove the existence of relaxation oscillations by using singular perturbation method and give an asymptotic expression of the period of relaxation oscillations. Finally, the parameter ranges which allow the relaxation oscillations in several scenarios are explored and displayed by conducting numerical simulations.

Integer-fractional decomposition and stability analysis of fractional-order nonlinear dynamic systems using homotopy singular perturbation method

Achieving a simplified model is a major issue in the field of fractional-order nonlinear systems, especially large-scale systems. So that in addition to simplifying the model, the outstanding features of the fractional-order modeling, such as memory feature, are preserved. This paper presented the homotopy singular perturbation method (HSPM) to reduce the complexity of the model and use the advantages of both models of the fractional order and the integer order. This method is a combination of the fractional-order singular perturbation method (FOSPM) and the homotopy perturbation method (HPM). Firstly, the FOSPM is developed for fractional-order nonlinear systems; then, a modification of the HPM is proposed. Finally, the HSPM is presented using a combination of these two methods. fractional-order nonlinear systems can be divided into two lower-order subsystems such as nonlinear or linear integer-order subsystem and linear fractional-order subsystem using this hybrid method. Convergence analysis of tracking error to zero is theoretically presented, and the effectiveness of the proposed method is also evaluated with two examples. Next, the number and location of equilibrium points are compared between the original system and the subsystems obtained from the proposed method. In the end, we show that the stability of fractional-order nonlinear system can be determined by investigating the stability of the subsystems using Theorem 3 and Lemma 2.

Singular Perturbation Analysis for Identification of Dynamic Behaviour and Stability of a Nonlinear Model of Long Term Progression of Diabetes Mellitus

There have been numerous attempts to model the progression of Diabetes Mellitus, which is a disease suffered by those with eating disorders with prevalence in the aged population. Models in the past have not been very successful in discovering the future development of the symptoms in a long term prediction. This is due to the fact that the state variables under consideration change in drastically different time scales, and the models that do not take careful account of this are not able to provide sufficiently accurate forecast that can be of satisfactory assistance to physicians taking care of their patients. In this work, we use the singular perturbation method to analyse a model of insulin and glucose interaction, incorporating beta cell dynamics and the pancreatic reserve, proposed by De Gaetano et al. in 2008. Different dynamic behaviour will be identified and numerical simulations will be carried out in support of our theoretical predictions.

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.

Singular Perturbation Analysis of a Coupled System Involving the Wave Equation

This article considers a system coupling an ordinary differential equation with a wave equation through its boundary data. The existence of a small parameter in the wave equation (as a factor multiplying the time derivative) suggests the idea of applying a singular perturbation method to get the stability of the full system by analyzing the stability of some appropriate subsystems given by the method. However, for infinite-dimensional systems, it is known that in some cases, this method does not work. Indeed, one cannot be sure of the stability of the full system even if the given subsystems are stable. In this article, we prove that the singular perturbation method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which is the first of this kind for systems involving the wave equation. Simulations are performed to show the applicability of our results.

Development of Analytical Solution for a Two-Phase Stefan Problem in Artificial Ground Freezing Using Singular Perturbation Theory

Abstract Artificial ground freezing (AGF) has historically been used to stabilize underground structure. Numerical methods generally require high computational power to be applicable in practice. Therefore, it is of interest to develop accurate and reliable analytical frameworks for minimizing computational cost. This paper proposes a singular perturbation solution for a two-phase Stefan problem that describes outward solidification in AGF. Specifically, the singular perturbation method separates two distinct temporal scales to capture the subcooling and freezing stages in the ground. The ground was considered as a porous medium with volume-averaged thermophysical properties. Further, Stefan number was assumed to be small, and effects of a few site-dependent parameters were investigated. The analytical solution was verified by numerical results and found to have similar conclusions yet with much lesser computational cost. Keywords: artificial ground freezing, Stefan-like problems, singular perturbation, porous media, outward solidification.

Solitary Waves and Periodic Waves in a Perturbed KdV Equation

In this paper, we consider a perturbed Korteweg–de Vries (KdV) equation with weak dissipation and Marangoni effects. Main attention is focused on the existence conditions of periodic and solitary wave solutions of the perturbed KdV equation. Based on bifurcation theory of dynamic system and geometric singular perturbation method, the parameter conditions and wave speed conditions for the existence of one periodic solution, one solitary solution and the coexistence of a solitary solution and infinite number of periodic solutions are given. By using Chebyshev criterion to analyze the ratio of Abelian integrals, the monotonicity of wave speed is proved, and the upper and lower bounds of wave speed are obtained.

A novel reduced i-propanol-n-butanol-ethanol (IBE)/diesel mechanism for engine combustion and emissions prediction

The direct application of i-propanol-n-butanol-ethanol (IBE) as an assuring alternative biofuel of internal combustion engines to get rid of the high recovery cost and extra energy consumption associated with n-butanol production has been gaining popularity, the research on performance and emissions of engines fueled with IBE are starting to be carried out. However, the available chemical reaction mechanism of IBE for engine combustion model is rarely reported. Therefore, a reduced mechanism named HT-IBE including 151 species and 775 reactions to track the combustion and emissions behavior of engine fueled with IBE/diesel blends was proposed in the present research. It was first meticulously developed based on the coupling of the reduced mechanisms of i-propanol, n-butanol, ethanol and diesel surrogates and the simplified NOx and soot formation models using combined reduction methods of directed relation graph with error propagation and sensitivity analysis (DRGEPSA), computational singular perturbation (CSP) and reaction path analysis (RPA) methods. To improve the predictive power of the mechanism, the sensitive reactions to ignition delay were identified and optimized based on sensitivity analysis. The mechanism was validated against ignition delay, laminar flame speed, and species mole fractions over a wide range of engine relevant conditions. The 3D engine simulation was further carried out using KIVA-3V2-CANTERA coupled with the mechanism to simulate the in-cylinder pressure, heat release rate, NOx and the soot emissions. Results show that the HT-IBE mechanism can reliably predict the combustion characteristics and emissions characteristics of IBE/diesel blends in the CI engine.

Multiple Positive Solutions for a Nonlocal Pde with Critical Sobolev-Hardy and Singular Nonlinearities Via Perturbation Method.

AbstractIn this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent$$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < $\begin{array}{} \displaystyle 2_s^* \end{array}$, where $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by$$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).

Singular Perturbation Method for Boundary Value and Optimal Problems to Power Factor Correction Converter Application

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Boundary Value Problem (BVP) and Optimal Problem. It is applied to two state switching power converters to provide rigorous justification of\ the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for BVP and optimal control problems. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

Invasion implies substitution in ecological communities with class-structured populations

Long-term evolution of quantitative traits is classically and usefully described as the directional change in phenotype due to the recurrent fixation of new mutations. A formal justification for such continual evolution ultimately relies on the “invasion implies substitution”-principle. Here, whenever a mutant allele causing a small phenotypic change can successfully invade a population, the ancestral (or wild-type) allele will be replaced, whereby fostering gradual phenotypic change if the process is repeated. It has been argued that this principle holds in a broad range of situations, including spatially and demographically structured populations experiencing frequency- and density-dependent selection under demographic and environmental fluctuations. However, prior studies have not been able to account for all aspects of population structure, leaving unsettled the conditions under which the “invasion implies substitution”-principle really holds. In this paper, we start by laying out a program to explore and clarify the generality of the “invasion implies substitution”-principle. Particular focus is given on finding an explicit and functionally constant representation of the selection gradient on a quantitative trait. Using geometric singular perturbation methods, we then show that the “invasion implies substitution”-principle generalizes to well-mixed and scalar-valued polymorphic multispecies ecological communities that are structured into finitely many demographic (or physiological) classes. The selection gradient is shown to be constant over the evolutionary timescale and that it depends only on the resident phenotype, individual growth-rates, population steady states and reproductive values, all of which are calculated from the resident dynamics. Our work contributes to the theoretical foundations of evolutionary ecology.

SECOND-ORDER STOCHASTIC VOLATILITY ASYMPTOTICS AND THE PRICING OF FOREIGN EXCHANGE DERIVATIVES

We consider models for the pricing of foreign exchange derivatives, where the underlying asset volatility as well as the one for the foreign exchange rate are stochastic. Under this framework, singular perturbation methods have been used to derive first-order approximations for European option prices. In this paper, based on a previous result for the calibration and pricing of single underlying options, we derive the second-order approximation pricing formula in the two-dimensional case and we apply it to the pricing of foreign exchange options.

Reduction and verification of models of surrogate fuels for RP-3 aviation kerosene coupled with NOx

The coupling of a semi-detailed chemical reaction model of three-component surrogate fuels for RP-3 aviation kerosene with NOx-related reactions in the GRI-Mech 3.0 methane mechanism is realized with three different methods. Two schemes are used in the reduction process while using the program ReaxRed to simplify the chemical reaction mechanisms coupled with NOx-related reactions. First, the revised-direct relation graph (Revised-DRG) method is used, providing an initial reduction. The result is, thereafter, applied to the computational singular perturbation (CSP) method (scheme 1) or to the path flux analysis (PFA) method (scheme 2) to enhance the reduction process, resulting in six simplified models. The comparison of several important parameters, such as ignition delay times (IDTs) between simplified models, initial models, and experimental data is realized to elect the best simplified model (109 components, 498 steps). Two methods are, therefore, developed to simulate NOx emission characteristics with the simplified models in a jet stirred combustion reactor (JSCR). The simulation results are, afterward, compared with the experimental results. As a result, the elected simplified model is also able to predict the NOx emission characteristics of RP-3 aviation kerosene within a specific range of conditions.

Dynamic behaviors of a predator–prey model with weak additive Allee effect on prey

This paper deals with a reaction–diffusion system modeling predator–prey interaction with additive Allee effect. We first analyze nonnegative constant equilibrium solutions and study the stability of the unique positive solution. Then we investigate the steady state bifurcation and Hopf bifurcation from the unique positive constant solution, respectively. The main tools used here include the bifurcation theory, the space decomposition and the implicit function theorem. Moreover, we describe the global structure from a simple eigenvalue. Finally, for the case d1=0, we prove the existence of far-from-equilibrium steady states with jump discontinuities. Moreover, by applying the singular perturbation method, we give a proof of the existence of large amplitude solutions when d1 is sufficiently small.

Maximum power extraction from fractional order doubly fed induction generator based wind turbines using homotopy singular perturbation method

In this paper, a novel control mechanism is presented for the control of variable speed wind turbine (VSWT) with a fractional-order doubly fed induction generator (DFIG) to extract the maximum power in region 2. Firstly, the homotopy singular perturbation method (HSPM) is proposed to simplify the fractional order nonlinear model, to reduce the long memory attribute, and to decrease the mechanical stress. This method is a composition of fractional order singular perturbation method (SPM) and modification of the homotopy perturbation method (HPM). Using this method, the nonlinear fractional order DFIG based VSWT is separated into two lower order subsystems, including nonlinear integer order subsystem and linear fractional order subsystem. Integer order modeling is applied to have the maximum wind power extraction and to increase the trajectory tracking speed, while the fractional order modeling is exerted to reduce the mechanical loads and to ensure better tracking. Then, a sliding mode controller based on HSPM is designed and applied to the original fractional order nonlinear system. Stability of the original system is evaluated by investigating the Lyapunov stability theorem for each subsystem. The controller performance designed based on HSPM is compared to the designed controller performance for fractional order nonlinear system decomposed using fractional order SPM, integer order nonlinear system decomposed by SPM, and integer order nonlinear system without simplification. The simulation results affirm the effectiveness of the controller design based on the proposed method.

Adaptive speed tracking controller for a brush-less DC motor using singular perturbation.

Abstract This work proposes a speed tracking controller for a brush-less DC (BLDC) motor in presence of plant parameters uncertainty and lack of current sensors. The designed controller is based on singular perturbation and adaptive control methods. Singular perturbation method is used to reduce the plant model order neglecting the current fast dynamics. Based on this reduced order model, an adaptive control law that has no dependency of the phase currents measurements, is formulated. The effectiveness of the proposed controller is demonstrated by numerical simulations.

PNP Models with the Same Positively Charged Valence

A steady-state Poisson-Nernst-Planck model with n ion species is studied under the assumption that n - 1 positively charged ion species have the same valence and there is only one negatively charged ion species. By re-scaling, this model can be viewed as a standard singularly perturbed system. Based on the explicit formulae for the solutions of its limit slow system and singular perturbation methods, the existence of the solutions is analyzed.

Integral control of stable nonlinear systems based on singular perturbations

One of the main issues related to integral control is windup, which occurs when, possibly due to a fault, the input signal u of the plant reaches a value outside the allowed input range U. This paper presents an integral controller with anti-windup, called saturating integrator, for a single-input single-output nonlinear plant having a curve of locally exponentially stable equilibrium points that correspond to constant inputs in U. A closed-loop system is formed by connecting the saturating integrator in feedback with the plant. The control objective is to make the output signal y of the plant track a constant reference r, while not allowing its input signal u to leave U. Using singular perturbation methods, we prove that, under reasonable assumptions, the equilibrium point of the closed-loop system is exponentially stable, with a “large” region of attraction. Moreover, when the state of the closed-loop system converges to this equilibrium point, then the tracking error tends to zero. A step-by-step procedure is presented to perform the closed-loop stability analysis, by finding separately a Lyapunov function for the reduced (slow) model and a Lyapunov function for the boundary-layer (fast) system. Afterwards, a Lyapunov function for the closed-loop system is built as a convex combination of the two previous ones, and an upper bound on the controller gain is found such that closed-loop stability is guaranteed. Finally, we show that if certain stronger conditions hold, then the domain of attraction of the stable equilibrium point of the closed-loop system can be made large by choosing a small controller gain.

Computational Singular Perturbation Method for Nonstandard Slow-Fast Systems

The computational singular perturbation (CSP) method is an algorithm which iteratively approximates slow manifolds and fast fibers in multiple-timescale dynamical systems. Since its inception due t...