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

Abstract

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.

1. Introduction

Under the background of more and more serious environmental problems and people’s increasing concerns about environmental protection, developing and efficiently using renewable energy has become a key topic of research [1]. Wind and solar power generation are quite affected by natural environment conditions, such as weather and light changes, thus revealing volatility, intermittence and randomness. In order to mitigate these issues and to ensure stable and safe operation of the power grid, the development of effective energy storage systems is crucial [2].

Energy storage technologies can be grouped into electrochemical, physical and thermal categories. The electrochemical storage technology is the most widely adopted, and it relies on LIBs as one of the most mature storage media [3]. Indeed, LIBs are used in many equipments due to their characteristics, namely long service life, high energy density, low self-discharge rate, and small size. However, to ensure efficient, safe and stable operation, accurate state estimation is a crucial part of the LIBs management system. The premise to develop accurate state estimation methods is the availability of high-precision battery models [4].

For an efficient battery management system, a high-precision model that fits the dynamic characteristics of the LIB is very important. The so-called equivalent circuit model approach has been widely used, since it yields good precision, has simple structure and allows easy implementation [5]. The standard equivalent circuit model is made of a series of resistors, capacitors and inductors. However, the electrochemical impedance spectrum of a pure capacitor is inconsistent with the actual LIB dynamics. As such, equivalent circuit models were proposed that include FO components, denoted as constant phase elements (CPE) [6], which have great influence on the model accuracy, and reveal a certain correlation with the SOC [7,8,9,10,11].

In addition to the model, the accuracy of the model parameters is critical [7,12,13]. Parameters’ identification methods for LIBs have been widely addressed in the literature [14,15,16]. For instance, schemes based on PSO algorithms were discussed in references [9,17,18], while methods based on the recursive least squares technique were investigated in other works [19,20,21]. The parameters’ identification method based on PSO has some limitations, such as population clustering caused by random initialization, lack of diversity, easy to fall in local optima, and slow convergence speed.

In this paper, an FIM that fits the LIB dynamics is set up based on the electrochemical impedance spectroscopy (EIS) technique. A new CAFPSO method is then proposed to accurately identify the FIM parameters. Chaotic initialization is used to improve the diversity of the initial population, and an adaptive fractional speed update law is adopted to increase the convergence rate of the algorithm and prevent it to fall into local optimal solutions [22]. The new CAFPSO significantly improves the accuracy of the FIM parameters’ identification. The accuracy and effectiveness of the approach are assessed by means of the RMSE and the MAE of the LIB terminals voltage, which are equal to 8.99 mV and 4.56 mV, thus showing excellent accuracy.

The paper is structured into five sections. Section 2 introduces the basics of FO calculus and the FO capacitor device. Section 3 addresses the mathematical derivation of the LIBs FIM. Section 4 presents the CAFPSO and assesses the accuracy of the algorithm. Finally, Section 5 concludes the paper.

2. Preliminary Concepts

Basic Concepts of FO Calculus

The tools of calculus are commonly used for the mathematical description of dynamical systems. The FO calculus was put forward by Leibnitz in 1695 as an extension of the integer calculus to any real or complex orders. In recent decades, it has been realized that real physical systems, namely those including diffusion phenomena, hysteresis, hydrodynamics, and other characteristics, are better described by the tools of FO calculus than by standard integer-order models. For LIBs, the FO calculus is able to accurately describe their internal electrochemical dynamical characteristics.

Let us denote by $D_t^{\alpha }$ the derivative $(\alpha >0)$ or integral $(\alpha <0)$ operators of order $\alpha \in \mathbb{R}$, with respect to t, where $t_0$ is the initial time instant. Herein, we will consider $t_0=0$ and $(\alpha >0)$ and use the simplified notation $D^{\alpha }$. Moreover, the Grünwald–Letnikov (GL) definition will be adopted. Therefore, given a function $x\left(t\right)$, we can write [23]:

$$\begin{array}{c}\hfill D^{\alpha }x\left(t\right)=\underset{\Delta T\to 0}{lim}\frac{1}{\Delta T^{\alpha }}\sum _{j=0}^{[t/\Delta T]}{(-1)}^j\left(\begin{array}{c}\alpha \\ j\end{array}\right)x(t-j\Delta T),\end{array}$$

(1)

where $\Delta T$ represents the sampling period, the operator $[·]$ stands for the integer part of the argument, and $\left(\begin{array}{c}\alpha \\ j\end{array}\right)$ corresponds to the generalization to real numbers of the Newton binomial, that is,

$$\begin{array}{c}\hfill \left(\begin{array}{c}\alpha \\ j\end{array}\right)=\frac{\Gamma (\alpha +1)}{\Gamma (j+1)·\Gamma (\alpha -j+1)},\end{array}$$

(2)

with $\Gamma \left(\alpha \right)$ denoting the Gamma function:

$$\begin{array}{c}\hfill \Gamma \left(\alpha \right)={\int }_0^{+\infty }{\xi }^{\alpha -1}{e}^{-\xi }d\xi .\end{array}$$

(3)

In the state-space form, for a linear time-invariant FO system, we have:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill D^{\alpha }x\left(t\right)& =Ax\left(t\right)+Bu\left(t\right)\hfill \\ \hfill y\left(t\right)& =Cx\left(t\right)+Du\left(t\right)\hfill \end{array}\right.\end{array}$$

(4)

3. The FO Modeling of LIBs

The EIS technique is often used to measure the AC impedance spectrum that characterizes the LIB dynamic response. Figure 1 (top) illustrates the impedance of a typical LIB using a Nyquist plot. The spectrum includes three main parts: (1) a low-frequency band in which the diffusion of solid-phase lithium-ions in the LIB electrodes occurs, (2) a middle-frequency band due to the charge transfer reaction and electrochemical double layer effect, and (3) a high-frequency band due to Ohmic resistance of inductive components. Integer-order equivalent circuit models use RC branches to model the mid-range band and ignore the characteristics of the low-frequency region. Moreover, as the EIS spectrum of an ideal capacitor is a semicircle in the Nyquist diagram, it is inconsistent with the ellipse-like curve actually observed. Thus, integer-order models cannot accurately describe the dynamic characteristics of LIBs. In this way, several works proposed using CPEs, instead of ideal capacitors, to establish an FIM that better describes the internal characteristics of LIBs. The impedance of a CPE is expressed as:

$$\begin{array}{c}\hfill Z\left(s\right)=\frac{1}{C{s}^{\alpha }},\end{array}$$

(5)

where C is related to the element capacitance.

Figure 1 (bottom) represents the FIM constructed to describe the dynamic characteristics of LIBs, where the circuit elements that model the regions (1)–(3) of the EIS are shown. In the model, the symbol I is the discharge current, OCV stands for the open circuit voltage, $V_0$ represents the battery terminals voltage, $R_0$ is the Ohmic internal resistance, $C_1$ and W are the CPEs, $R_1$ is the resistance on the RC parallel, $V_1$ corresponds to the voltage across the RC branch, and $V_2$ is the Warburg-like element voltage. Correspondingly, we use the W element to simulate the LIB characteristics in the low-frequency band, the RC parallel formed by a CPE and $R_1$ to model the characteristics in the medium-frequency band, and $R_0$ to simulate the high-frequency band.

As the curve representing the low-frequency band in the Nyquist diagram is always a 45° straight line, the FO of W is set to $0.5$. The curve corresponding to the middle-frequency band changes with the SOC [24], so we express the FO ${\alpha }_1$ of the CPE in the RC parallel as a polynomial function of the SOC, yielding:

$$\begin{array}{c}\hfill {\alpha }_1\left(SOC\right)=\sum _{i=0}^7{b}_{i}SO{C}^i\left(t\right).\end{array}$$

(6)

The OCV-SOC relationship is usually a nonlinear function, which we expressed as:

$$\begin{array}{c}\hfill U_{OCV}=\sum _{i=0,i\ne 1}^6{a}_{i}SO{C}^i\left(t\right).\end{array}$$

(7)

The FIM of the LIB can then be established based on the Kirchhoff’s voltage and current laws, yielding:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& D^{{\alpha }_1\left(SOC\right)}{x}_1\left(t\right)=-\frac{V_1\left(t\right)}{R_1{C}_1}+\frac{1}{C_1}u\left(t\right),\hfill \\ & D^{0.5}{x}_2\left(t\right)=\frac{1}{W}u\left(t\right),\hfill \\ & D^1{x}_3\left(t\right)=-\frac{\eta }{Q_n}u\left(t\right),\hfill \\ & y\left(t\right)=OCV\left(x_3\left(t\right)\right)-x_1\left(t\right)-x_2\left(t\right)-R_{0}u\left(t\right),\hfill \end{array}\right.\end{array}$$

(8)

where the function $u\left(t\right)$ stands for the input current, with a negative (positive) signal for charging (discharging), the vector $x\left(t\right)={[V_1\left(t\right),V_2\left(t\right),SOC\left(t\right)]}^T$ is the state, and the function $y\left(t\right)$ denotes the output voltage. Moreover, the symbol $Q_n$ represents the nominal capacity of the LIB, which is expressed in Ampere-hour (Ah), and the symbol $\eta$ corresponds to the Coulomb efficiency.

Based on expression (4), the FO system given by (8) can be represented in the discrete-time, k, ($k\ge 1$), $\alpha =[{\alpha }_1,0.5,1]$ as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill x(k+1)& =[{(\Delta T)}^{\alpha }A+diag\left(\alpha \right)I]x\left(k\right)\hfill \\ & -\sum _{j=2}^{L+1}{(-1)}^j\left(\begin{array}{c}\alpha \\ j\end{array}\right)x(k+1-j)\hfill \\ & +{(\Delta T)}^{\alpha }Bu\left(k\right),\hfill \\ \hfill y\left(k\right)& =OCV\left(x_3\left(k\right)\right)-x_1\left(k\right)-x_2\left(k\right)-R_{0}u\left(k\right),\hfill \end{array}\right.\end{array}$$

(9)

where

$$\begin{array}{c}\hfill {(\Delta T)}^{\alpha }=diag({(\Delta T)}^{{\alpha }_1(SOC)},{(\Delta T)}^{0.5},\Delta T),\end{array}$$

(10)

$$\begin{array}{c}\hfill \left(\begin{array}{c}\alpha \\ j\end{array}\right)=diag\left(\left(\begin{array}{c}{\alpha }_1\left(SOC\right)\\ j\end{array}\right),\left(\begin{array}{c}0.5\\ j\end{array}\right),\left(\begin{array}{c}1\\ j\end{array}\right)\right),\end{array}$$

(11)

$$\begin{array}{c}\hfill A=\left[\begin{array}{ccc}-\frac{1}{R_1{C}_1}& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],B=\left[\begin{array}{c}\frac{1}{C_1}\\ \frac{1}{W}\\ -\frac{\eta }{Q_n}\end{array}\right].\end{array}$$

(12)

4. Parameters’ Identification and FIM Validation

As seen in Figure 1, the FIM has unknown parameters, and obtaining their values is demanding. Therefore, it is crucial to accurately identify the LIM parameters by means of suitable methods. Herein, we propose a new CAFPSO method and assess its effectiveness.

The PSO is a simple and easy to implement algorithm inspired on birds’ predation behavior. However, the traditional PSO has some drawbacks, such as the slow rate of convergence and it easy to fall into local optimal solutions, which leads to considerable errors in the identification process.

The initial position of the population is crucial in the performance of PSO. The logistic chaotic map can be adopted to generate an evenly distributed population, thus contributing to improve the performance of the algorithm. The logistic chaotic map is expressed as:

$$\begin{array}{c}\hfill {\mathrm{x}}_{\mathrm{k}+1}=\mu {\mathrm{x}}_{\mathrm{k}}\left(1-{\mathrm{x}}_{\mathrm{k}}\right),\end{array}$$

(13)

where k is the iteration number, and $\mu \in (0,4]$.

For logistic chaotic mapping, when $\mu =4$, the system is chaotic, the distribution uniformity of the mapping reaches its peak, and the number of particle swarm is 300. Here, we adopt $\mu =4$ and 300 iterations to generate the logistic chaotic sequence. The values are mapped to particle individuals to perform the initialization of the population, according to the formula:

$$\begin{array}{c}\hfill \chi ={\chi }_{lb}+\left({\chi }_{ub}-{\chi }_{lb}\right){\chi }_{k+1},\end{array}$$

(14)

where ${\chi }_{lb},{\chi }_{ub}$ are the upper and lower limits of each individual in each dimension, and $\chi$ is the mapped individual.

On the other hand, the AFPSO was introduced as a novel technique for controlling the convergence rate of the PSO algorithm that uses the concepts of FO calculus. It was shown that the FO directly influences the speed of convergence of the PSO. In order to avoid the algorithm falling into local optima and to improve search efficiency, an FO velocity adaptation law is adopted:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill D^q\left({\nu }_i^{k+1}\right)& =C_1{R}_1(P_{bes{t}_i}^k-{\chi }_i^k)+C_2{R}_2(G_{bes{t}_i}^k-{\chi }_i^k),\hfill \\ \hfill {\chi }_i^{k+1}& ={\chi }_i^k+{\nu }_i^k,\hfill \end{array}\right.\end{array}$$

(15)

Then, using the GL derivative (1) and $r=4$ terms, one has:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\nu }_i^{k+1}& =q{\nu }_i^{k+1}+\frac{1}{2}q(1-q){\nu }_i^{k-1}\hfill \\ & +\frac{1}{6}q(1-q)(2-q){\nu }_i^{k-2}\hfill \\ & +\frac{1}{24}q(1-q)(2-q){\nu }_i^{k-3}\hfill \\ & +C_1{R}_1(P_{bes{t}_i}^k-{\chi }_i^k)+C_2{R}_2(G_{bes{t}_i}^k-{\chi }_i^k),\hfill \\ \hfill {\chi }_i^{k+1}& ={\chi }_i^k+{\nu }_i^k,\hfill \end{array}\right.\end{array}$$

(16)

where q is adaptively updated by:

$$\begin{array}{c}\hfill q=0.8-0.5\frac{k}{K},\end{array}$$

(17)

with k denoting the current iteration and K denoting the total number of iterations. For good performance, $q\in [0.3,0.8]$ is chosen.

The FIM parameters to be identified are encapsulated into the vector $\varphi =[b_o,b_1,b_2,b_3,$$b_4,b_5,b_6,b_7,R_0,R_1,C_1,W]$. The parameters’ identification objective is to find the parameters values that minimize the fitness function, $f(·)$, that represents the error between the measured and the predicted voltages, $V_0\left(k\right)$ and ${\widehat{V}}_0\left(k\right)$, respectively, meaning:

$$\begin{array}{c}\hfill minf(·)=min\sum _{k=1}^M{[V_0\left(k\right)-{\widehat{V}}_0\left(k\right)]}^2,\end{array}$$

(18)

where the symbol M denotes the total number of the sampling points. Figure 2 illustrates schematically the CAFPSO algorithm.

The RMSE and the MAE metrics are adopted to assess the accuracy of the CAFPSO, which are computed as:

$$\begin{array}{ccc}\hfill RMSE& =& \sqrt{\frac{1}{M}\sum _{k=1}^M{[V_0\left(k\right)-{\widehat{V}}_0\left(k\right)]}^2},\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill MAE& =& \frac{1}{M}\sum _{k=1}^M\mid V_0\left(k\right)-{\widehat{V}}_0\left(k\right)\mid .\hfill \end{array}$$

(20)

For assessing the effectiveness of the method, we use data of voltage and current under LIB Dynamic Stress Test (DST) conditions. The nonlinear relationship between OCV and SOC was fitted by a 6th-order polynomial based on low-current OCV test data.

Table 1. The identified coefficients of the OCV-SOC polynomial.

${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{a}}_6$
3.159	2.499	−9.197	17.860	−16.717	8.259	−1.816

summarizes the identified polynomial coefficients and

Table 2. The result of the parameters’ identification with the CAFPSO.

${\mathit{b}}_0$	${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_3$	${\mathit{b}}_4$	${\mathit{b}}_5$
0.9932	−1.7075	1.6735	−0.4257	−1.2296	3.0521
$b_6$	$b_7$	$R_0$	$R_1$	$C_1$	W
−2.2960	0.6695	0.0721	0.3685	1977	976.211

lists the parameters’ values obtained with the new CAFPSO. In order check the effectiveness of the CAFPSO, we compare the voltages estimated when using the parameters identified by the CAFPSO, the PSO and the AFPSO. Figure 3 and Figure 4 depict the obtained voltage and the voltage absolute error for 25 °C. From Figure 3, we verify that the parameters identified by the CAFPSO yield better results than the ones identified by both the PSO and AFPSO alternatives. From Figure 4 and

Table 3. The values of the RMSE and MAE metrics of the LIB terminals voltage with the parameters’ identification by the CAFPSO, PSO and AFPSO.

	PSO	AFPSO	CAFPSO
RMSE (mV)	21.4	11.6	8.99
MAE (mV)	16.8	6.95	4.56

, we verify that the RMSE by the CAFPSO, PSO and AFPSO are 8.99 mV, 11.6 mV and 21.4 mV, respectively, while the MAE yields 4.56 mV, 6.95 mV and 16.8 mV. Therefore, the identified parameters lead to good accuracy, and the proposed FIM and CAFPSO are reliable.

5. Conclusions

A new CAFPSO algorithm was derived to obtain accurate parameters’ identification of LIBs models. A FIM was set up for modeling the LIB based on the EIS technique. A CAFPSO was then derived based on the logistic chaotic mapping method and used to initialize the particles’ population in order to increase diversity. A speed updating law with adaptive FO was adopted to adjust the particle express delivery and position. The effectiveness of the CAFPSO was verified by comparing the LIB terminals voltage obtained with the parameters identified by the CAFPSO, the PSO and the AFPSO. Accuracy improvements of $22.5\%$ and $34.4\%$, and $57.9\%$ and $72.8\%$, in the RMSE and MAE, respectively, were obtained by the CAFPSO relative to the PSO and AFPSO. Future research will consider the temperature in the process of parameters’ identification, since it was found that the accuracy of the FIM depends on this variable.