Fractional-order Equivalent Circuit Model Research Articles

Uneven Usage Battery State of Health Estimation via Fractional-Order Equivalent Circuit Model and AutoML Fusion

To accurately predict the State of Health (SOH) of lithium-ion batteries under the continuously changing charging and discharging conditions in practical applications, this study proposes a hybrid modeling approach that integrates a Fractional Order Equivalent Circuit Model (F-ECM) with the AutoGluon automatic machine learning framework. By leveraging Electrochemical Impedance Spectroscopy (EIS) to capture battery frequency response characteristics, F-ECM accurately fits EIS data to extract detailed internal state parameters. The integration of AutoGluon automates the machine learning process, enhancing the precision of SOH predictions. Through testing and analysis on real battery datasets, this method has demonstrated superior prediction precision and computational efficiency compared to existing mainstream modeling approaches. Specifically, the hybrid method achieved a Root Mean Square Error (RMSE) of 2.12% and a Mean Absolute Error (MAE) of 1.67%. This study presents a highly accurate, interpretable, and adaptable predictive framework for lithium-ion battery health assessment, offering valuable insights for battery health management system development.

Dynamic Fractional-Order Model of Proton Exchange Membrane Fuel Cell System for Sustainability Improvement

The proton exchange membrane fuel cell (PEMFC) stands at the forefront of advancing energy sustainability. Effective monitoring, control, diagnosis, and prognosis are crucial for optimizing the PEMFC system’s sustainability, necessitating a dynamic model that can capture the transient response of the PEMFC. This paper uses a dynamic fractional-order model to describe the behaviors of a stationary micro combined heat and power (mCHP) PEMFC stack. Based on the fractional-order equivalent circuit model, the applied model accurately represents the electrochemical impedance spectroscopy (EIS) and the dynamic voltage response under transient conditions. The applied model is validated through experiments on an mCHP PEMFC stack under various fault conditions. The EIS data is analyzed under different current densities and various fault conditions, including the stoichiometry of the anode and cathode, the stack temperature, and the relative humidity. The dynamic voltage response of the applied model shows good correspondence with experimental results in both phase and amplitude, thereby affirming the method’s precision and solidifying its role as a reliable tool for enhancing the sustainability and operational efficiency of PEMFC systems.

A Comparative Study of Fractional-Order Models for Supercapacitors in Electric Vehicles

Fractional-order models display many advantages compared with integer-order equivalent circuit models for modeling energy storage system, such as more precision, fewer parameters. This paper carries out a comparative study of fractional-order equivalent circuit models for modeling supercapacitor. These models are fractional-order RC model, fractional-order classic model, fractional-order RRCW model, fractional-order dynamic model, fractional-order transmission line model. An experiment load platform is built to test the supercapacitor for collecting the feature data. To be fair, the parameters of all models are optimized to achieve their best fitting accuracy by using the genetic algorithm. Accuracy and complexity two indices are considered to evaluate the above five models based on different test cycles. Simulation results show that the fractional-order RRCW model has the ideally comprehensive performance.

Accurate Electromagnetic Force Analysis of Offshore Wind Power Transformer Windings Based on Fractional Order Lumped Parameter Model

Accurate calculation of the electromagnetic force distribution of transformer windings under different loads and fault conditions is of great significance for transformer maintenance, condition evaluation and life prediction. Due to the influence of offshore wind power systems, offshore wind power transformers have high harmonic content and large changes in load rates, which can easily cause the coil destabilization, winding deformation or even damage because of the uneven distribution of the electromagnetic force. To improve the accuracy of electromagnetic force calculation, this paper proposes a fractional order numerical method. First, a three-dimensional axisymmetric transformer model and a symmetrical lumped parameter equivalent circuit model are established, respectively, based on field-circuit coupling. Second, the fractional order approximation of circuit components is realized by using the improved Oustaloup filter. In the fractional order model, the transformer is replaced by the lumped parameter equivalent circuit model. Third, as in the calculation process for integer order electromagnetic force, the integer order current has a large error, and the current waveform does not match the actual power frequency. The fractional order current and electromagnetic force at the 0.9 order are closer to the rated value. Finally, the effects of different load rates, three-phase short circuits and harmonic conditions are studied with the fractional order model. Compared with the traditional integer order finite element electromagnetic model, the fractional order equivalent circuit model established in this paper is more accurate and suitable for electromagnetic force calculation. The proposed method is significant for the structural design and state detection of transformers and also could be applied in the analysis of other dry-type transformers.

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.

Embedded real-time fractional-order equivalent circuit model for internal resistance estimation of lithium-ion cells

This paper introduces a new on-board capable method for model-based estimation of the internal resistance of lithium-ion cells. The method relates to a fractional-order electrical Equivalent-Circuit-Model (ECM) consisting of ohmic resistances, ZARC and Warburg elements. Based on a Begin-of-Life (BOL) model parameterization by means of Electrochemical-Impedance-Spectroscopy (EIS), a subset of the model parameters is tracked offline over cell aging to generate End-of-Life (EOL) model parameters, using time domain measurements and a nonlinear optimization routine. The model parameters over lifetime are trained in quadratic polynomials for on-board application on the Battery-Management-System (BMS). The on-board aging estimation uses filtering algorithms and feedback control of the difference between the measured and calculated cell voltages during dynamical driving. An embedded real-time implementation, combining float- and fixed-point arithmetic, is carried out for a state-of-the-art BMS. Validation tests using a vehicle with an aged battery containing 108 cells confirm fast and stable convergence of all cell’s State-of-Health-Resistance (SOHR). The SOHR estimation convergence is reached within at most one hour of dynamical driving with an estimation error under 3.5 %. Furthermore, a proposed model-based Open-Circuit-Voltage (OCV) estimation is validated with relaxation experiments, confirming an estimation error of less than 1 mV.

A physics-based fractional-order equivalent circuit model for time and frequency-domain applications in lithium-ion batteries

Equivalent circuit models (ECMs) remain the most popular choice for online applications in lithium-ion batteries because of their simpler parameterization and lower computational requirements in comparison to electrochemical models. Nevertheless, standard ECMs lack physical insight and fail to accurately reproduce cell behavior under a wide range of operating conditions. For this reason, the development of physics-informed ECMs becomes essential so as to provide a better description of the physical processes while maintaining a reduced computational complexity. In this article, we propose a novel physics-based ECM derived directly from an electrochemical model, so that there is a clear correlation between circuit states and internal battery states, as well as circuit and physical parameters. The proposed model yields an RMS error below 1.46 mV for cell voltage, 0.28% for the surface concentration in the active material particles, 0.6% for the electrode-averaged electrolyte concentration and 0.32 mV for the charge-transfer overpotentials. Another key feature of this model is the relationship between circuit parameters and those identified in frequency-domain tests, which allows us to characterize and validate the model experimentally. We understand that the presented model constitutes an alternative to standard ECMs as well as electrochemical models as it combines advantageous characteristics from both of them.

A framework for battery temperature estimation based on fractional electro-thermal coupling model

The temperature significantly affects the performance and safety of the battery. Therefore, this paper presents a novel framework for the estimation of the battery temperature based on a fractional-order electro-thermal coupling model. Due to the nonlinearity, coupling, and time varying parameters of lithium-ion batteries, the fractional-order equivalent circuit model is established using the fractional-order differential theory. Then it couples with the thermal model to establish the fractional order electro-thermal coupling model of lithium battery. Based on the model, the hybrid pulse power characteristic test method and the electro-thermal balance experiment are used to identify the parameters of the fractional order electro-thermal coupling model of lithium batteries. Finally, the fractional-order electro-thermal coupling model is used to estimate battery voltage, state of charge, and temperature under the conditions of the hybrid pulse power characteristic and the world transient vehicle cycle. The results show good consistency and responsiveness with the experiment under steady and dynamic conditions, and compared with the ETM, the characteristics of small error and low error rates of the F-ETM are introduced. Therefore, the fractional electrothermal coupling model of lithium batteries has good accuracy and applicability.

Improved Parameter Identification for Lithium-Ion Batteries Based on Complex-Order Beetle Swarm Optimization Algorithm

With the aim of increasing the model accuracy of lithium-ion batteries (LIBs), this paper presents a complex-order beetle swarm optimization (CBSO) method, which employs complex-order (CO) operator concepts and mutation into the traditional beetle swarm optimization (BSO). Firstly, a fractional-order equivalent circuit model of LIBs is established based on electrochemical impedance spectroscopy (EIS). Secondly, the CBSO is used for model parameters' identification, and the model accuracy is verified by simulation experiments. The root-mean-square error (RMSE) and maximum absolute error (MAE) optimization metrics show that the model accuracy with CBSO is superior when compared with the fractional-order BSO.

Frequency domain parametric estimation of fractional order impedance models for Li-ion batteries

The impedance of a Li-ion battery contains information about its state of charge (SOC), state of health (SOH) and remaining useful life (RUL). Commonly, electrochemical impedance spectroscopy (EIS) is used as a nonparametric data-driven technique for estimating this impedance from current and voltage measurements. In this article, however, we propose a consistent parametric estimation method based on a fractional order equivalent circuit model (ECM) of the battery impedance. Contrary to the nonparametric impedance estimate, which is only defined at the discrete set of excited frequencies, the parametric estimate can be evaluated in every frequency of the frequency band of interest. Moreover, we are not limited to a single sine or multisine excitation signal. Instead, any persistently exciting signal, like for example a noise excitation signal, will suffice. The parametric estimation method is first validated on simulations and then applied to measurements of commercial Samsung 48X cells. For now, only batteries in rest, i.e. at a constant SOC after relaxation, are considered.

Dynamic analysis and experiment of chaotic circuit of non-homogeneous fractional memristor with bias voltage source

A physical memristor has an asymmetric tight hysteresis loop. In order to simulate the asymmetric tight hysteresis curve of the physical memristor more conveniently, a fractional-order diode bridge memristor model with a bias voltage source is proposed in this paper, which can continuously regulate the hysteresis loop. Firstly, based on fractional calculus theory, a fractional order model of a diode bridge memristor with a bias voltage source is established, and its electrical characteristics are analyzed. Secondly, by integrating it with the Jerk chaotic circuit, a non-homogeneous fractional order memristor chaotic circuit model with a bias voltage source is established, and the influence of bias voltage on its system dynamic behavior is studied. Once again, a fractional-order equivalent circuit model is built in PSpice and validated through circuit simulation. The experimental results are basically consistent with the numerical simulation results. Finally, the experiments on the circuit are completed in LabVIEW to validate the correctness and feasibility of the theoretical analysis. The results indicate that the fractional order memristor with bias voltage source can continuously obtain asymmetric tight hysteresis loop by adjusting the voltage of the bias voltage source. As the bias power supply voltage changes, the non-homogeneous fractional order memristor chaotic system exhibits that the period doubling bifurcation turns into chaos due to the symmetry breaking.

Time-domain system identification of Li-ion batteries from non-zero initial conditions

Electrochemical Impedance Spectroscopy (EIS) allows characterizing electrochemical behavior of Lithium-ion batteries. However, it requires the battery to be at a relaxed state which is quite time-consuming during data acquisition process at different State-Of-Charge (SOC) levels. A time relaxation is usually observed of about 1h at each SOC level. In the prospect to reduce the measurement time, this paper presents a time-domain identification method using non-zero initial conditions of a fractional-order equivalent circuit model (FO-ECM). A two-stage iterative algorithm is developed. It estimates at one stage the system free response, due to the past initialization, and the system forced response due to the input/output signal. It uses, at the second stage, an output error model to estimate model parameters. The algorithm is iterated until convergence. The proposed identification algorithm is applied to identify a Li-ion battery FO-ECM using experimental data.

Improved coyote optimization algorithm for parameter estimation of lithium-ion batteries

This paper studies the parameter estimation of fractional order equivalent circuit model of lithium-ion batteries. Since intelligent optimization algorithms can achieve parameters with high accuracy by transforming the parameter estimation into optimization problem, coyote optimization algorithm is taken in this paper by modifying two key steps so as to improve the accuracy and convergence speed. Firstly, tent chaotic map is introduced to avoid falling into local optimum and enhance population diversity. Secondly, dual strategy learning is employed to improve the searching ability, accuracy and convergence speed. Non-parametric statistical significance is tested by 6 benchmark functions with the comparison of other 5 optimization algorithms. Furthermore, the proposed algorithm is applied to identify the fractional order model of the Samsung ICR18650 (2600 mAh) and compared with conventional coyote optimization algorithm and particle swarm algorithm, which declared the excellence in identification accuracy.

Parameter identification method for the variable order fractional-order equivalent model of lithium-ion battery

The equivalent model structure, fractional order-value, and basic circuit parameters in a fractional-order equivalent circuit model (FECM) of a battery jointly determine the description ability of the FECM. For a certain model structure, the fractional order-value determines the basic form of frequency domain impedance characteristics of the model. However, in practical applications, the determination of the fractional order-value lacks clear guidelines. In this paper, the orders of FECM are taken as parameters to be identified, and the coevolutionary particle swarm optimization algorithm is used to synchronously identify the parameters and order-values of equivalent model from a complete set of load profile test data. Under the same algorithm conditions, the state description capabilities of FECM are compared based on different fractional order-values. Furthermore, the reasonable fractional order-value interval is determined, which provides a reference value for the order-value selection in the fixed-order FECM. Test results show that for the two fractional order-values used in the 2-FRC equivalent model, when one is a fixed value, the other fluctuates in a moderate range in most cases. According to this principle, a fixed-order FECM is used to identify the battery parameters. On the basis of not losing the description accuracy of FECM, it reduces the computational complexity caused by fractional order operation.

Variable Fractional-Order Equivalent Circuit Model for Lithium-Ion Battery via Chaotic Adaptive Fractional Particle Swarm Optimization Method

A variable fractional-order equivalent circuit model is proposed to accurately describe the dynamic characteristics of lithium-ion batteries (LIBs). Firstly, a fractional impedance model (FIM) is established, such that the fractional-order (FO) is a polynomial function of the LIB state of charge (SOC). Then, a chaotic adaptive fractional particle swarm optimization (CAFPSO) method is derived to identify the parameters of the FIM. Experiments reveal the reliability of the novel approach through the root-mean-squared error (RMSE) and the mean absolute error (MAE) of the LIB terminals voltage, yielding the values 8.99 mV and 4.56 mV, respectively. This translates into accuracy improvements of 22.5% and 34.4% for the particle swarm optimization (PSO) algorithm and 57.9% and 72.8% for the adaptive fractional particle swarm optimization (AFPSO) algorithm, respectively.

Structural identifiability of impedance spectroscopy fractional-order equivalent circuit models with two constant phase elements

Structural identifiability analysis of fractional-order equivalent circuit models (FO-ECMs) obtained through electrochemical impedance spectroscopy (EIS) remains as a challenging problem. Aside from practical challenges such as measurement noise and the selection of excitation signals, no widely accepted analytical or numerical proof addressing the structural identifiability of impedance spectroscopy FO-ECMs exists. Through the use of coefficient mapping technique, this paper proposes novel computationally-efficient algebraic equations for numerical structural identifiability analysis of a widely used FO-ECM with Grünwald–Letnikov fractional derivative approximation and two constant phase elements (CPEs) including the Warburg term. The effects of the length of the data and sampling time on the proposed method for the structural identifiability analysis of the two-CPE FO-ECMs are discussed along with two examples and a discussion of the results.

Diagnosis of proton exchange membrane fuel cell system based on adaptive neural fuzzy inference system and electrochemical impedance spectroscopy

A new diagnostic method based on adaptive neural fuzzy inference system (ANFIS) and electrochemical impedance spectroscopy (EIS) is proposed for the proton exchange membrane fuel cell (PEMFC) system. Firstly, a new parameter identification method that combines genetic algorithm (GA) and Levenberg–Marquardt (LM) algorithm is proposed to identify the fractional-order equivalent circuit model (ECM), in which the anode impedance, cathode impedance, and mass transfer are all considered. This new method allows better exploitation of the EIS diagrams, and the internal relationships between the fault conditions and the ECM parameters are thoroughly analyzed according to it. Then, based on these relationships, a new diagnostic algorithm based on k-means clustering and ANFIS is designed to precisely identify several faults that can occur in the PEMFC, such as membrane flooding, drying, and mass transfer fault. Finally, the effectiveness of this method is demonstrated experimentally through the exploitation of EIS data under different faults and operating conditions of the PEMFC.

State of power estimation of lithium-ion battery based on fractional-order equivalent circuit model

State of power (SOP) is an important parameter to characterize the power performance of lithium-ion battery. Different from State of Charge (SOC), SOP estimation assumes that the battery is operated in extreme working condition, so higher accuracy and robustness are required for the battery model. In this study, the fractional-order equivalent circuit model is adopted to estimate SOP, which takes SOC, voltage, and current of the battery as constraints with the fractional-order calculus. Through the estimation results analysis under the FUDS and DST driving cycles, the general guidelines of the SOP constraint process are summarized: the discharge SOP is sequentially constrained by current, voltage, and SOC, and when the current changes frequently, the hybrid constraint is easy to be generated; the constraint process of charge SOP is opposite. Besides, novel experiments are designed separately to verify the SOP estimation results under SOC, voltage, and current constraints. The experimental results show that the maximum error of SOP estimation results is 1.34%, which proves that the SOP estimation based on the fractional-order model provides high estimation accuracy.

A Simplified Fractional Order Modeling and Parameter Identification for Lithium-Ion Batteries

Abstract With the widespread development of new energy, the study of power lithium-ion batteries (LIBs) has broad prospects and great academic significance. The model and parameters are two essential prerequisites for LIB state estimation, which are used to provide a guarantee for the secure and convenient handling of LIBs. To obtain the reliable model and parameters, a simplified fractional order equivalent circuit model (FO-ECM) with high precision is presented in this article. The dynamic external electrical characteristic of LIBs is represented by the one-order FO-ECM, and then, the FO-ECM parameters are identified by the combination of Grunwald–Letnikov (G-L) definition-based factional order numerical calculation and noise compensation-based forgetting factor recursive least squares (FFRLS) method. The simplified FO-ECM can better characterize the nonlinear dynamic behaviors of LIBs, and the G-L definition-based FO-FFRLS algorithm can maintain good accuracy in the parameter estimation process. The results show that the simplified FO-ECM can improve the modeling precision and parameter identification performance compared with the common integer-order ECM in different test cycles.

A novel online identification algorithm of lithium‐ion battery parameters and model order based on a fractional order model

A fractional order equivalent circuit model with variable order can better reflect the internal reaction mechanism of a lithium-ion battery, however, it is not easy to simultaneously identify the parameters and order of the fractional order model online. In this paper, based on the electrochemical impedance spectroscopy of the lithium-ion battery, the FOM and its discretization are analysed in detail, and a fractional order repeated prediction recursive least square (FORPRLS) method is proposed, the approximate error of fractional derivative is corrected by the repeated identification process with changing the order, so that the parameters and order of the fractional order model can be identified online and accurately at the same time. For the second-order RC equivalent circuit model and fractional order model, different online identification algorithms including the forgetting factor recursive least square and FORPRLS are carried out under the dynamic stress test experiment, and the predicted terminal voltages are compared with the actual measured terminal voltages to judge the identification accuracy of these algorithms. The experimental results verify that the FORPRLS algorithm is superior to the FFRLS algorithm, and the identified order can better match the reaction process of the lithium-ion battery.