Fractional-order Model Parameters Research Articles

Fast acquisition method of battery electrochemical impedance spectra based on impedance fragments

Electrochemical impedance spectroscopy (EIS) is often used for battery characterization and monitoring. However, there are still difficulties in acquiring battery impedance spectra in large quantities and wide frequency bands in practical applications. The lower the frequency range of acquisition, the longer the measurement time spent, and the higher the frequency range of measurement, the more expensive the measurement equipment, and these problems seriously limit the wide application of impedance information. To address this problem, this study proposes a method for fast acquisition of electrochemical impedance spectra of lithium-ion batteries based on impedance fragments. Firstly, impedance segments at specific frequencies in the EIS are selected as features based on the distribution of relaxation times (DRT). Then, a Long Short-Term Memory (LSTM) neural network model is used to reconstruct the complete impedance spectrum, and a population optimization algorithm is used to optimize the hyperparameters in the model. The validation results indicate that the reconstructed RMSE error is only 14.26 mΩ, corresponding to a MAPE of 1.65 %, while the measurement time is reduced to 27.01 % of the original measurement time. Finally, the performance of this method in identifying fractional-order model parameters is further discussed.

Analyzing the dynamics of fractional spatio-temporal $ \mathrm{SEIR} $ epidemic model

<p>In this work, we present a comprehensive analysis of the spatio-temporal $ \mathrm{SEIR} $ epidemic model of fractional order. The infection dynamics in the proposed fractional order model (FOM) are described by a system of partial differential equations (PDEs) within a time-fractional order and diffusion operator in one-dimensional space, considering that the total population is split into four compartments: Susceptible, exposed, infected, and recovered individuals denoted as $ \mathrm{S} $, $ \mathrm{E} $, $ \mathrm{I} $ and $ \mathrm{R} $, respectively. Our contributions commence by establishing the existence and uniqueness of positively bounded solutions for the proposed FOM. Moreover, we determined all equilibrium points (EPs) and investigated their local stability based on the basic reproduction number (BRN) $ \mathcal{R}_{0} $, which is calculated by the next-generation matrix (NGM) method. Additionally, we demonstrated global stability using an appropriate Lyapunov function with fractional LaSalle's invariance principle (LIP). Sensitivity analysis of the FOM parameters was discussed to identify the most critical parameters by which the volume of disease propagation can be measured. The theoretical findings were corroborated by numerical simulations of solutions that are displayed in 3D and 2D graphs. Graphical simulations highlight the effect of vaccination on infection severity. Changing the fractional order $ \alpha $ in the proposed FOM has an influence on the speed of convergence to the steady state as a result of the memory effect. Furthermore, vaccination emerges as an effective strategy for disease control.</p>

Fractional-Order Modeling of Piezoelectric Actuators with Coupled Hysteresis and Creep Effects

A novel fractional-order model, incorporating coupled hysteresis and creep effects, is proposed for typical piezoelectric actuators in this study. Throughout the actuation process, various nonlinear behaviors such as piezoelectric hysteresis, non-local memory, peak transition, and creep nonlinearity are accurately characterized by the model. Offering a simpler structure and superior tracking performance compared to conventional models, the proposed fractional-order model parameters are identified using a method that integrates actuator dynamics and employs the particle swarm optimization algorithm. Experimental validation on a piezoelectric actuation platform confirms the model’s superior accuracy and simplified structure, contributing to a deeper understanding of piezoelectric actuation mechanisms and providing an efficient modeling tool for enhanced system performance.

A totally coupled multi time-scale framework containing full parameters online identification and SOC real-time estimation of lithium-ion battery based on a fractional order model

Fractional order model (FOM) is one of the most promising choices for lithium-ion battery simulation and internal states estimation due to its high modeling accuracy. However, online identifying the order and parameters of FOM is extremely tough, which also makes it difficult to be applied in complex and variable operating conditions. To address this issue, this paper proposes a totally coupled multi time-scale framework that contains full parameters online identification of FOM and state-of-charge (SOC) real-time estimation of lithium-ion battery. Firstly, a novel global optimization algorithm named improved particle swarm optimization is designed to update the order of FOM on macro-time scale with low computational burden. On the basis of the determined order, the rest parameters of FOM are further identified by forgetting factor recursive least square algorithm on micro-time scale. With those online identified parameters, adaptive extended Kalman filter is finally adopted to track battery SOC in every instant. The estimation accuracy, adaptability to different operating temperatures and battery aging, and robustness ability against uncertain initialization of the developed method is verified under Federal Urban Driving Schedule tests. The corresponding results demonstrate that the proposed method can realize accurate full parameters online identification of FOM and precise SOC real-time estimation against different temperatures, battery aging states and initialization. Using the developed method, the mean absolute error (MAE) and root mean square error (RMSE) of SOC estimation of the fresh battery under temperature ranging from 0 °C to 50 °C can be limited below 0.9 % and 1.1 %, respectively. Even if the SOC-OCV relationship is not timely updated, those two indexes of the aged battery achieved by the proposed method can still be controlled within 3 % under temperature ranging from 0 °C to 50 °C. Moreover, the comparison with other typical FOM-based estimation methods is also conducted, whose results indicate that the developed method has apparently superior comprehensive performance on estimation accuracy and adaptation.

On full-life-cycle SOC estimation for lithium batteries by a variable structure based fractional-order extended state observer

Accurate SOC estimation of lithium batteries are crucial for the efficient operation of new energy storage systems. During the ageing of the battery, structure and parameters of the battery model, especially internal resistance, may change, which has a particularly significant impact on the accuracy of the model. For this reason, this paper proposes a SOC estimation method based on the extended state observer of the variable structure fractional order model. Firstly, an adaptive method for the structure and parameters of fractional order model through distribution of relaxation times (DRT) is proposed on full-cycle-life of lithium battery. The DRT is extracted from the Electrochemical Impedance Spectroscopy (EIS) of the lithium battery. The order and the initial parameters of the fractional order model of the lithium battery is determined by the characteristics of DRT during the ageing process of the lithium battery. Adaptive adjustment of model is realized by parameter identification combining with time domain data. Then, a fractional-order extended state observer is proposed to estimate SOC by treating internal resistance as an extended state, thus realizing online estimation of internal resistance uncertainty. The Lyapunov stability analysis proves that the estimation error of the observer is uniformly ultimately bounded. Finally, the experimental simulation analysis shows that the accuracy of the second-order model is significantly improved compared with the first-order model, and the accuracy improvement of the third-order model is limited compared with the second-order model. The MAE of the proposed algorithm is as low as 0.73%.

Numerical study on radiative MHD flow of viscoelastic fluids with distributed-order and variable-order space fractional operators

The magnetohydrodynamic (MHD) flow has been concerned widely for its widespread adoption in the field of astrophysics, electronics and many other industries over the years. The purpose of this article is to introduce the variable and distributed order space fractional models to characterize the MHD flow and heat transfer of heterogeneous viscoelastic fluids in a parallel plates. Based on the central difference approximation of Riesz space fractional derivative, the Crank–Nicolson difference scheme for the governing equations is established, and the effectiveness of the algorithm is verified by two numerical examples. We examine the effects of fractional-order model parameters on the velocity and temperature, our investigation indicates that for the constant fractional model, the larger the fractional order parameter, the smaller the velocity and temperature. The variable space fractional method can be used to describe dynamic behavior with time and space dependence, while the distributed space fractional model can describe various phenomena in which the number of differential orders varies over a certain range, characterizing their complex processes over space, and it is also more suitable for simulating the fluid flow and thermal behavior of complex viscoelastic magnetic fluid.

Identification of fractional-order equivalent circuit model of lithium-ion battery for improving estimation of state of charge

Accurate estimation of State of Charge (SOC) based on equivalent circuit model (ECM) of Lithium-ion batteries (LIBs) is an important research topic. Recent research has found that fractional-order model (FOM), as one kind of equivalent circuit model, can provide a better description of LIBs dynamics than the conventional integer-order ECMs. However, it is difficult to directly identify an FOM online. In this paper, a synergy of Beetle Antennae Search and Recursive Least Squares (BAS-RLS) is proposed to identify the parameters of an FOM for LIBs. Specifically, the BAS is adopted to determine the fractional order, while the remaining parameters of FOM are estimated via the well-established RLS algorithm. A comparison study will demonstrate that this method offers a similar modeling accuracy as the Particle Swarm Optimization (PSO), and the online estimated model is more accurate than offline estimated models. Based on the identified FOM, this article then explores the battery SOC estimation from two aspects: initial value effect of the SOC and the comparison of multiple estimation algorithms, thus verifying the advantage of the proposed method for LIBs modeling and SOC estimation.

Long-memory recursive prediction error method for identification of continuous-time fractional models

This paper deals with recursive continuous-time system identification using fractional-order models. Long-memory recursive prediction error method is proposed for recursive estimation of all parameters of fractional-order models. When differentiation orders are assumed known, least squares and prediction error methods, being direct extensions to fractional-order models of the classic methods used for integer-order models, are compared to our new method, the long-memory recursive prediction error method. Given the long-memory property of fractional models, Monte Carlo simulations prove the efficiency of our proposed algorithm. Then, when the differentiation orders are unknown, two-stage algorithms are necessary for both parameter and differentiation-order estimation. The performances of the new proposed recursive algorithm are studied through Monte Carlo simulations. Finally, the proposed algorithm is validated on a biological example where heat transfers in lungs are modeled by using thermal two-port network formalism with fractional models.

Lung cancer dynamics using fractional order impedance modeling on a mimicked lung tumor setup

IntroductionAs pulmonary dysfunctions are prospective factors for developing cancer, efforts are needed to solve the limitations regarding applications in lung cancer. Fractional order respiratory impedance models can be indicative of lung cancer dynamics and tissue heterogeneity. ObjectiveThe purpose of this study is to investigate how the existence of a tumorous tissue in the lung modifies the parameters of the proposed models. The first use of a prototype forced oscillations technique (FOT) device in a mimicked lung tumor setup is investigated by comparing and interpreting the experimental findings. MethodsThe fractional order model parameters are determined for the mechanical properties of the healthy and tumorous lung. Two protocols have been performed for a mimicked lung tumor setup in a laboratory environment. A low frequency evaluation of respiratory impedance model and nonlinearity index were assessed using the forced oscillations technique. ResultsThe viscoelastic properties of the lung tissue change, results being mirrored in the respiratory impedance assessment via FOT. The results demonstrate significant differences among the mimicked healthy and tumor measurements, (p-values <0.05) for impedance values and also for heterogeneity index. However, there was no significant difference in lung function before and after immersing the mimicked lung in water or saline solution, denoting no structural changes. ConclusionSimulation tests comparing the changes in impedance support the research hypothesis. The impedance frequency response is effective in non-invasive identification of respiratory tissue abnormalities in tumorous lung, analyzed with appropriate fractional models.

Estimation of state of charge of lithium battery based on parameter identification of fractional order model

Accurate estimation of state of charge of lithium battery is one of the important performance parameters for safe and reliable operation of lithium battery. A fractional order second-order Thevenin equivalent circuit model was proposed by based on the improvement of the traditional Thevenin equivalent circuit model for accurately estimating the state of charge of lithium-ion battery. In order to overcome the shortcomings of the least square method easily enter into local convergence or even unable to converge, an adaptive genetic algorithm is proposed to identify the parameters of lithium battery model, and global parameter identification is carried out to improve the convergence of the algorithm. Matlab simulation shows that the parameters of fractional order model of the second-order Thevenin equivalent circuit identified by adaptive genetic algorithm are better than those of integer order model identified by least square method. Combined with extended Kalman filter, the estimation of state of charge accuracy control is realized, with the accuracy error being within 1.61%.

Novel Time Method of Identification of Fractional Model Parameters of Supercapacitor

Accurate dynamic models of supercapacitors (SCs) are a basis for the design, control and exploitation of the hybrid energy storage systems for electric vehicles. This paper concerns a fractional model of SC impedance, based on the Cole–Cole equation describing relaxation in electric double layer. This article provides a new method of identifying the parameters of fractional order model of SC impedance, performed without disconnecting the SC module from the energy storage system. The test drive for this purpose needs only the respect a few simple recommendations. The article presents the conditions of the mentioned test drive that will ensure the frequency spectra of the recorded signals lying in the bandwidth necessary for the correct identification of the model parameters. These parameters are determined by means of the Nelder–Mead simplex optimization method. The results of the identification described by the time method coincide with those obtained in the frequency domain. It has been shown in the last part of the article that the real energy losses in these systems significantly exceed the losses determined only on the basis of the nominal capacity and series equivalent resistance (ESR), to which the SC catalogue data are usually limited. This paper also provides an auxiliary frequency criterion for the selection of SCs intended for energy storage systems of electric vehicles.

Design and Implementation of an Optimized Artificial Human Eardrum Model

This paper introduces a fractional-order eardrum Type-II model, which is derived using fractional calculus to reduce the number of elements compared to its integer-order counterpart. The proposed fractional-order model parameters are extracted and compared using five meta-heuristic optimization techniques. The CMOS implementation of the model is performed using the Design Kit of the Austria Mikro Systeme (AMS) 0.35 $$\upmu {\hbox {m}}$$ CMOS process, while the simulations have been performed using the Cadence IC design suite.

Frequency and time domain modelling and online state of charge monitoring for ultracapacitors

Ultracapacitors are common power sources for new energy vehicles with the advantage of providing high current and power. Modelling and state of charge (SOC) estimation of ultracapacitors are still a research focus. In this study, a fractional order model (FOM), which is derived from the analysis of electrochemical impedance spectroscopy, is established for ultracapacitors and validated in frequency and time domains. After analyzing characteristics of an ultracapacitor and its open circuit voltage, the established FOM is combined with a fractional order Unscented Kalman filter to realize SOC estimation for ultracapacitors. Results show that the proposed method can estimate SOC more accurately than traditional SOC methods. Furthermore, an online joint estimation method for SOC and FOM parameters is proposed and validated by a hardware-in-the-loop platform, indicating that the FOM can be used for onboard applications.

Fractional-order Modelling and Parameter Identification of Electrical Coils

Abstract The accurate modelling of an electrical coil over a wide range of frequency is the keystone for a precise modelling of an electrical machine. As a consequence of copper losses, eddy-current losses and hysteresis losses; electrical coils with conductive ferromagnetic core show different behaviour from that of an ideal coil. Throughout this paper, dynamic modelling and performance analysis of conventional as well as fractional-order models of an electrical coil with an interchangeable core are achieved. Measurement results are acquired through an integration between Matlab and the high-speed measurement system LTT24. In order to assess the accuracy of these models, simulation results are compared with experimental results whereas unknown parameters are identified through an optimization process that is based on the method of least squares. It is known that the parameters of fractional-order model (Lα , α, Cβ , β) can not be measured directly. Therefore, the paper proposes a possibility based on system analysis to derive these parameters (indirect measurement) from the parameters of the classical model. A frequency band beyond the self-resonant frequency of the electrical coil is explored, thus the parasitic capacitance between coils windings must be considered as an important part of the equivalent circuit. The dependency of model parameters on frequency due to skin-effect is also examined.

Rates and Effects of Local Minima on Fractional-Order Circuit Model Parameters Extracted from Supercapacitor Discharging Using Least Squares Optimization

Optimization routines are widely used to numerically determine a set of model parameters that best fit collected experimental data. One recent application of these methods is to extract the fractional-order circuit model parameters that accurately characterize the transient behavior of discharging supercapacitors. However, the variability that these methods introduce to the extracted model parameters must be understood to determine if changes in model parameters are artifacts of the optimization routine or are representative of physical changes in the device under study. In this work, the variability of supercapacitor fractional-order model parameters is quantified when extracted using a nonlinear least squares optimization applied to simulated data (with 0.1–3% noise) and experimental data of their discharging behavior. These results indicate that the local minima problem occurs at a rate of $$<1\%$$ for this application, though this problem can be overcome using at least 10 independent executions of the solver to each dataset. The variability of the model parameters using 1000 executions of the solver with 1 and 10 iterates is quantified for both simulated and experimental data, with the 10 iterates showing at least an order of magnitude reduction in the range of returned parameters.

Fractional-Order Impedance Modeling and Parameter Identification of an Electrical Coil with Interchangeable Core

The accurate modelling of an electrical coil over a wide range of frequency is essential for a precise modelling of an electrical machine. As a consequence of copper losses, eddy-current losses and hysteresis losses; electrical coils with conductive ferromagnetic core show different behaviour from that of an ideal coil. Throughout this paper, dynamic modelling and parameter identification of fractional-order models of an electrical coil with interchangeable core are achieved. In order to assess the accuracy of these models, simulation results are compared with experimental results whereas unknown parameters are identified through an optimization process that is based on the method of least squares. It is well known that the parameters of fractional-order model can not be measured directly. Therefore, the paper proposes a possibility based on system analysis to derive these parameters from the parameters of the classical model.

Parameter Sensitivity Analysis for Fractional-Order Modeling of Lithium-Ion Batteries

This paper presents a novel-fractional-order lithium-ion battery model that is suitable for use in embedded applications. The proposed model uses fractional calculus with an improved Oustaloup approximation method to describe all the internal battery dynamic behaviors. The fractional-order model parameters, such as equivalent circuit component coefficients and fractional-order values, are identified by a genetic algorithm. A modeling parameters sensitivity study using the statistical Multi-Parameter Sensitivity Analysis (MPSA) method is then performed and discussed in detail. Through the analysis, the dynamic effects of parameters on the model output performance are obtained. It has been found out from the analysis that the fractional-order values and their corresponding internal dynamics have different degrees of impact on model outputs. Thus, they are considered as crucial parameters to accurately describe a battery’s dynamic voltage responses. To experimentally verify the accuracy of developed fractional-order model and evaluate its performance, the experimental tests are conducted with a hybrid pulse test and a dynamic stress test (DST) on two different types of lithium-ion batteries. The results demonstrate the accuracy and usefulness of the proposed fractional-order model on battery dynamic behavior prediction.

Fractional-order modeling of permanent magnet synchronous motor speed servo system

A fractional-order modeling approach for a permanent magnet synchronous motor speed servo system is proposed applying a method combining electromagnetic part modeling and mechanical part modeling. Based on the proposed fractional-order model and system identification scheme, system identification experiments are performed on the electromagnetic part and the mechanical part of the permanent magnet synchronous motor speed servo system, respectively. The fractional-order model parameters of these two parts are identified with these experimental results, and the fractional-order model of the permanent magnet synchronous motor speed servo system is integrated from these two parts. Simulations and experiments in open-loop and closed-loop are performed based on the obtained fractional-order model and integer-order model. The advantage of the proposed fractional-order model for the permanent magnet synchronous motor speed servo system is demonstrated by the simulation and experimental results.

Robust model predictive control of uncertain fractional systems: a thermal application

In this study, a robust model predictive control of fractional-order systems with real uncertain parameters is presented. The Grunwald–Letnikov's method is used as an internal model to predict the plant future dynamic behaviour. This method consists in replacing the non-integer derivation operator of the adopted system representation by a discrete approximation. Therefore a robust fractional model predictive control is developed based on an uncertain fractional model. Moreover, the output deviation approach is adopted to design the j-step ahead output predictor, and the corresponding control law is obtained by the resolution of a min–max optimisation problem that takes into account the uncertainties of the fractional-order model parameters and under input constraints. The efficiency and the performances of the proposed predictive controller are illustrated with practical results of a thermal system.

Estimation of the Order and Parameters of a Fractional Order Model From a Noisy Step Response Data1

This paper deals with integral based methods to estimate the order and parameters of simple fractional order models from the extracted noisy step response data of a process. This data can be obtained from both open-loop and closed-loop tests. Numerical simulation results are presented to verify the robustness of these proposed methods in the presence of the measurement noise.