Parameter Identification for Lithium-Ion Battery Based on Hybrid Genetic–Fractional Beetle Swarm Optimization Method

Abstract

This paper proposes a fractional order (FO) impedance model for lithium-ion batteries and a method for model parameter identification. The model is established based on electrochemical impedance spectroscopy (EIS). A new hybrid genetic–fractional beetle swarm optimization (HGA-FBSO) scheme is derived for parameter identification, which combines the advantages of genetic algorithms (GA) and beetle swarm optimization (BSO). The approach leads to an equivalent circuit model being able to describe accurately the dynamic behavior of the lithium-ion battery. Experimental results illustrate the effectiveness of the proposed method, yielding voltage estimation root-mean-squared error (RMSE) of 10.5 mV and mean absolute error (MAE) of 0.6058%. This corresponds to accuracy improvements of 32.26% and 7.89% for the RMSE, and 43.83% and 13.67% for the MAE, when comparing the results of the new approach to those obtained with the GA and the FBSO methods, respectively.

1. Introduction

Electric vehicles (EVs) are gradually replacing traditional fuel vehicles in order to achieve energy saving and carbon emission reduction [1,2]. The power batteries (PBs) are the energy source of EVs. Many types of PBs have been used to power EVs, such as nickel-cadmium, lead-acid, and nickel-metal hydride batteries. In contrast with these, lithium-ion batteries (LIBs) do not suffer from memory effects and are superior in terms of self-discharge rate, cycle life, and environmental friendliness [3,4]. Therefore, LIBs have become the preferred PBs for EVs.

The PBs present strong time-varying behavior, high dependence with ambient temperature and nonlinear dynamics [5]. Therefore, high-precision and robust state estimation become crucial to actively manage PBs. Improving the dynamic modeling of PBs for high accuracy and low computational burden is important for state estimation [6,7]. The equivalent circuit models (ECMs) are composed of electronic components, including resistors, capacitors and inductors, and have been proposed for battery monitoring and management, due to their fidelity in many applications and tractable computational load [8]. However, the double-layer characteristic observed in the mid-frequency region of the EIS response is not consistent with pure capacitance element models [9]. The misfitting has been solved by adopting constant phase elements (CPE). Indeed, the use of ECMs with CPEs leads to FO models (FOMs) of LIBs, which have been widely used in the energy management of batteries [10,11,12,13,14,15].

The accuracy of any LIB model depends not only on the model structure, but also on the effectiveness of the algorithm adopted for model parameter identification (PI) [16,17,18,19,20]. Several methods for PI have been investigated [21,22,23]. For instance, a particle swarm optimization (PSO) algorithm was discussed in [24,25,26], while a recursive least squares method was proposed in [27,28,29]. However, classical single optimization algorithms often suffer from poor stability and tend to fall into local optima. In contrast, hybrid algorithms can combine the advantages of different classical schemes, thereby yielding solutions with higher efficiency and quality [30,31]. For example, a coupling hybrid adaptive particle swarm optimization-simulated annealing (HAPSO-SA) method was adopted in [32], and a new PI method combining a real-time variable forgetting factor recursive least squares (VFFRLS) and an adaptive extended Kalman filter (AEKF) was proposed in [33].

In this paper, a FO impedance model (FIM) of LIBs is established based on EIS. A new HGA-FBSO method is suggested to improve the accuracy of the FIM. The algorithm combines the advantages of the GA and the BSO methods, mitigating the issues related to poor stability and tendency to reach local optima that characterize the classical single optimization algorithms. The accuracy and effectiveness of the new method are verified in the perspective of the RMSE and MAE of the voltage estimation, which are shown to be better that 10.5 mV and 0.6058%, thus revealing excellent accuracy.

The paper is organized as follows: Section 2 introduces some tools of the FO calculus and the FO capacitor element. Section 3 presents the mathematical derivation of the FIM for LIBs. Section 4 parameterizes the battery model and assesses its accuracy and effectiveness. Finally, Section 5 presents the conclusions of the study.

2. Preliminaries

2.1. Fractional-Order Calculus

The FO calculus deals with the generalization of the traditional calculus to non-integer orders [34,35]. There are several definitions of FO integro-differential operators, being the Grünwald–Letnikov (GL), the Caputo (C) and the Riemann–Liouville (RL) ones among the most used. Generally speaking, the RL definition is commonly used in theoretical analysis. The C formulation is well suited for the description and discussion of initial value problems of FO differential equations. The GL definition is often used in problems that require discretization. Herein, the GL definition is adopted due to its straightforward computational implementation. The GL operator of a function $x\left(t\right)$ is:

$$\begin{array}{ccc}\hfill D^{q}x\left(t\right)& =& \underset{T\to 0}{lim}\frac{1}{T^q}\times {\displaystyle \sum _{j=0}^{[t/T]}}{(-1)}^j\langle q,j\rangle x(t-jT),\hfill \end{array}$$

where $D^q$ corresponds to the FO operator, $q\in R$ is the integro-differential order (with $q>0$ and $q<0$ corresponding to derivative and integral, respectively, and $q=0$ yielding $D^q=1$). The parameter T is the sampling time, $[t/T]$ represents the memory length, and $\langle q,j\rangle$ is the Newton binomial coefficient, given by:

$$\begin{array}{c}\hfill \langle q,j\rangle =\frac{\Gamma (q+1)}{\Gamma (j+1)\Gamma (q-j+1)},\end{array}$$

(1)

where $\Gamma (·)$ stands for the Gamma function.

Often, the state space equation of a FO system can be represented by:

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

(2)

where $x\left(t\right)$, $u\left(t\right)$ and $y\left(t\right)$ are the state, input and output vectors, respectively, $A,B,C$ and D are coefficients matrices, and $\alpha =[{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n]$ is the vector of differentiation orders. Meanwhile, the system (2) can be discretized in the time-domain based on the GL definition. For $k\ge 1$ we get:

$$\begin{array}{ccc}\hfill x(k+1)& =& [T^{\alpha }A+diag\left(\alpha \right)I]x\left(k\right)\hfill \\ & -& {\displaystyle \sum _{j=2}^{L+1}}{(-1)}^j\frac{\Gamma (\alpha +1)}{\Gamma (j+1)\Gamma (\alpha -j+1)}x(k+1-j)+T^{\alpha }Bu\left(k\right),\hfill \end{array}$$

(3)

and, for $k=0$, it results:

$$\begin{array}{c}\hfill x(k+1)=[T^{\alpha }A+diag\left(\alpha \right)]x\left(k\right)+T^{\alpha }Bu\left(k\right).\end{array}$$

(4)

2.2. Fractional-Order Capacitor

The relationship between voltage and current for a conventional (integer-order) capacitor is:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& i\left(t\right)=C\frac{du\left(t\right)}{dt},t\ge 0,\hfill \\ & \frac{U\left(s\right)}{I\left(s\right)}=\frac{1}{Cs},\hfill \end{array}\right.\end{array}$$

(5)

where C represents the capacitance, $u\left(t\right)$ is a step (or a quasi-step) voltage applied across the terminals of the capacitor, and $i\left(t\right)$ is the current flowing through the element. In the Laplace domain we have $U\left(s\right)=L\left[u\right(t\left)\right],I\left(s\right)=L\left[i\right(t\left)\right]$, where $L[·]$ and s represent the Laplace transform and variable, respectively.

The U-I relationship for the FO capacitor is [36]:

$$\begin{array}{c}\left\{\begin{array}{c}i\left(t\right)=C_{\varphi }\frac{d^{\alpha }u\left(t\right)}{d{t}^{\alpha }},t\ge 0,\hfill \\ \frac{U\left(s\right)}{I\left(s\right)}=\frac{1}{C_{\varphi }{s}^{\alpha }}.\hfill \end{array}\right.\hfill \end{array}$$

(6)

where $C_{\varphi }$ is a constant related to the capacitance, and $\alpha$ represents the fractional order. Further, the concept of pseudo-capacitance was proposed, being represented by $C_{\alpha }$. Therefore, the impedance of a FO capacitor is:

$$\begin{array}{c}\hfill Z_{C_{\alpha },\alpha }\left(s\right)=\frac{U\left(s\right)}{I\left(s\right)}=\frac{1}{C_{\alpha }{s}^{\alpha }},\end{array}$$

(7)

where $C_{\alpha }$ represents a normalized capacitance. As such, the capacitance of FO capacitors is frequency-dependent, being given by:

$$\begin{array}{c}\hfill C_{\varphi }=\frac{C_{\alpha }}{{\omega }^{1-\alpha }}.\end{array}$$

(8)

3. Fractional-Order Modeling of Libs

The frequency-impedance characteristic test is used to measure the impedance of the PB in a given frequency range. This corresponds to the EIS response, which is a useful tool to represent the dynamic characteristics of batteries.

The classical ECMs used for battery modeling include the Rint, Thévenin, and dual polarization (DP) models. The Rint has simple structure, but its accuracy is often insufficient. Compared with the Rint, the Thévenin model adopts a resistor-capacitor (RC) branch to describe the polarization characteristics of the battery, while the DP model uses two RC branches, since polarization is classified into concentration and electrochemical polarization. Contrasting with the DP model, the FIM includes a Warburg element to describe the low-frequency characteristics of the battery, which results into higher accuracy. In addition, the FIM with CPE can represent more accurately the internal characteristics of the battery, when compared to integer-order models. Therefore, a FIM is adopted herein to model the LIB.

A common representation of a battery in terms of EIS response is shown in the upper part of Figure 1. Three regions can be identified: (1) straight-line low-frequency region; (2) semi-oval mid-frequency part; (3) high-frequency zone of inductance effect. The low-frequency region originates from the diffusion of solid-phase lithium-ions within the battery electrodes. Such effect can be captured by a Warburg-like CPE. The mid-frequency semi-ellipse-like part is explained by the electrochemical double layer and charge transfer reaction. Indeed, as polarization is classified into concentration and electrochemical types, a series of resistance in parallel with CPE are used for the sake of accuracy. The high-frequency tail of the curve is interpreted in the light of the Ohmic resistance of inductive elements, such as current cables and collectors. It should be noted that the battery Ohmic internal resistance, $R_0$, corresponds to the intersection of the impedance spectrum with the real axis.

Based on the above discussion, a FIM of LIB is drawn in the bottom part of Figure 1. The relevant circuit elements corresponding to the three regions of the EIS curve are represented. In the model, OCV denotes open circuit voltage, I represents the discharge current, $V_0$ stands for the terminal voltage, $C_1,C_2$ and W are the CPE coefficients, $R_1$ and $R_2$ are the resistances on the two RC branches, $V_1$ and $V_2$ represent the voltages on the RC branches, and $V_3$ is the voltage on the Warburg-like element.

Based on the Kirchhoff’s current and voltage laws, the battery model can be established and the governing equations of the FIM can be given by:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill D^{{\alpha }_1}{x}_1\left(t\right)& =-\frac{V_1\left(t\right)}{R_1{C}_1}+\frac{1}{C_1}u\left(t\right),\hfill \\ \hfill D^{{\alpha }_2}{x}_2\left(t\right)& =-\frac{V_2\left(t\right)}{R_2{C}_2}+\frac{1}{C_2}u\left(t\right),\hfill \\ \hfill D^{{\alpha }_3}{x}_3\left(t\right)& =\frac{1}{W}u\left(t\right),\hfill \\ \hfill D^1{x}_4\left(t\right)& =-\frac{\eta }{Q_n}u\left(t\right),\hfill \\ \hfill y\left(t\right)& =OCV\left(x_4\left(t\right)\right)-x_1\left(t\right)-x_2\left(t\right)-x_3\left(t\right)-R_{0}u\left(t\right),\hfill \end{array}\right.\end{array}$$

(9)

where $u\left(t\right)$ represents the input current, being positive (negative) for discharging (charging) operations. The state vector is $x\left(t\right)={[V_1\left(t\right),V_2\left(t\right),V_3\left(t\right),SOC\left(t\right)]}^T$, and $y\left(t\right)$ stands for the output voltage. Moreover, $Q_n$ represents the nominal capacity of the battery expressed in Ampere-hour (Ah) and $\eta$ is the Coulomb efficiency. Based on (3), the FO system (9) in discrete-time, with time step k, can be expressed as ($k\ge 1$):

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill x(k+1)& =[{(\Delta T)}^{\alpha }A+diag\left(\alpha \right)I]x\left(k\right)\hfill \\ & -{\displaystyle \sum _{j=2}^{L+1}}{(-1)}^j\left(\begin{array}{c}\alpha \\ j\end{array}\right)x(k+1-j)\hfill \\ & +{(\Delta T)}^{\alpha }Bu\left(k\right),\hfill \\ \hfill y\left(k\right)& =OCV\left(x_4\left(k\right)\right)-x_1\left(k\right)-x_2\left(k\right)\hfill \\ & -x_3\left(k\right)-R_{0}u\left(k\right).\hfill \end{array}\right.\end{array}$$

(10)

where

$$\begin{array}{c}\hfill {(\Delta T)}^{\alpha }=diag({(\Delta T)}^{{\alpha }_1},{(\Delta T)}^{{\alpha }_2},{(\Delta T)}^{{\alpha }_3},\Delta T),\end{array}$$

(11)

$$\left(\begin{array}{c}\alpha \\ j\end{array}\right)=diag(\left(\begin{array}{c}{\alpha }_1\\ j\end{array}\right),\left(\begin{array}{c}{\alpha }_2\\ j\end{array}\right),\left(\begin{array}{c}{\alpha }_3\\ j\end{array}\right),\left(\begin{array}{c}1\\ j\end{array}\right)),$$

(12)

$$A=\left[\begin{array}{cccc}-\frac{1}{R_1{C}_1}& 0& 0& 0\\ 0& -\frac{1}{R_2{C}_2}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],B=\left[\begin{array}{c}\frac{1}{C_1}\\ \frac{1}{C_2}\\ \frac{1}{W}\\ -\frac{\eta }{Q_n}\end{array}\right].$$

(13)

4. Model Parameter Identification and Validation

The battery models are often time-varying and nonlinear. Moreover, as seen in Figure 1, the battery FIM has several parameters, which are difficult to obtain directly due to the complexity of the electrochemical reactions involved. In the follow-up, we present a procedure to parameterize the FIM.

The main voltage and current data of the battery INR 18650-20R are used. The nominal capacity of the battery is 2000 mAh, and the cut-off voltage is 4.2 V. The current and voltage data of Dynamic Stress Test (DST) operating conditions are used for PI.

Hybrid optimization methods, namely, the genetic algorithm-least squares (GA-LS) [37] and the hybrid genetic algorithm/particle swarm optimization (HGAPSO) [38], have been adopted to identify FOM parameters, including the fractional differentiation orders. The hybrid optimization algorithms can combine the advantages of different methods to improve efficiency and quality of the solutions, revealing higher effectiveness when compared with single optimization algorithms. Indeed, no single optimization algorithm can be used to solve all optimization problems, and there are always limitations associated with single optimization algorithms. For example, the BSO is an extension of the beetle antennae search (BAS) and PSO algorithms. However, the performance of the BAS in dealing with high-dimensional functions is not satisfactory, and the iterative result is very dependent on the initial position of the beetle. Inspired by the swarm optimization approach, we proposed further improvements to the BAS by expanding from individual to group optimization, yielding the BSO algorithm. It is also well known that the distribution of the initial population significantly impacts the convergence of an intelligent optimization algorithm. For example, in the BSO, as the initial population is often generated randomly, the individuals result unevenly distributed, leading to insufficient diversity in the entire search process. To solve the problem, we embed the crossover and mutation operations in the BSO to help preserving population diversity. Based on the above discussion, the novel hybrid method HGA-FBSO is proposed herein to accomplish model PI.

The HGA-FBSO combines the main advantages of the GA and FBSO, and leads to more accurate optimization results. The GA relies on the crossover and mutation operations to help preserving swarm diversity and, thus, better reducing the risk of falling into local optima. A meta-heuristic algorithm is an iterative approach that guides a subordinate heuristic for solving a problem, while using a combination of randomization and local search. The core of a meta-heuristic algorithm is exploration and exploitation. Exploration means that it tries to explore the entire search space, because the optimal solution may exist anywhere. Exploitation means that it uses effective information as much as possible to search from the initial toward the optimal solution. Meta-heuristic algorithms became very popular due to their stability and flexibility, as well as to their ability to avoid local optima. The BSO is a new meta-heuristic algorithm, and can enhance the performance of swarm optimization through beetle foraging principles. The BSO is based on the PSO and BAS algorithms and incorporates an incremental factor generated by the beetle search to perform a local search [39,40]. Therefore, the BSO can alleviate the issues of poor stability and solutions falling into local optima. Further, the FBSO has memory properties that can capture the past behavior of particles for modeling the remembering and forgetting of the particles past behavior. The framework of the HGA-FBSO is sketched in Figure 2. The FO velocity and position update formulas are:

$$\begin{array}{ccc}\hfill D^{\beta }\left({\nu }_i^{k+1}\right)& =& c_1{r}_1(P_{bes{t}_i}^k-x_i^k)+c_2{r}_2(G_{bes{t}_i}^k-x_i^k),\hfill \\ \hfill x_i^{k+1}& =& x_i^k+\lambda {\nu }_i^k+(1-\lambda ){\xi }_i^k,\hfill \end{array}$$

(14)

where $P_{bes{t}_i}^k$ is the best position found so far, $G_{bes{t}_i}^k$ is the best position of the swarm, $c_1$ and $c_2$ represent the learning factors, $r_1$ and $r_2$ denote random numbers between $[0,1]$, $\lambda$ is the proportional constant, and ${\xi }_i^k$ stands for the increment factor.

Using the $GL$ derivative given in Equation (1), one has:

$$\begin{array}{ccc}\hfill {\nu }_i^{k+1}& =& \beta {\nu }_i^k+\frac{1}{2}\beta (1-\beta ){\nu }_i^{k-1}+\frac{1}{6}\beta (1-\beta )(2-\beta ){\nu }_i^{k-2}\hfill \\ & +& \frac{1}{24}\beta (1-\beta )(2-\beta )(3-\beta ){\nu }_i^{k-3}\hfill \\ & +& \cdots +c_1{r}_1(P_{bes{t}_i}^k-x_i^k)+c_2{r}_2(G_{bes{t}_i}^k-x_i^k),\hfill \\ \hfill x_i^{k+1}& =& x_i^k+\lambda {\nu }_i^k+(1-\lambda ){\xi }_i^k,\hfill \end{array}$$

(15)

with ${\xi }_i^k$ updated by

$$\begin{array}{ccc}\hfill {\xi }_i^{k+1}& =& {\delta }^k{\nu }_i^k·sign(f\left(x_{ir}^k\right)-f\left(x_{il}^k\right)),\hfill \\ \hfill {\delta }^{k+1}& =& \eta ·{\delta }^k,\hfill \end{array}$$

(16)

where ${\delta }^k$ represents the step size of the k-th iteration, sign(·) represents a signum function, $f\left(x_{ir}^k\right)-f\left(x_{il}^k\right)$ is the scent intensity difference between the right antennae position, $x_{ir}^k$, and the left antennae position, $x_{il}^k$, of beetle i, and $\eta$ is a constant to be adjusted by the designer. Here we set $\eta =0.95$. The search behavior of the right and left antennas are:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill x_{ir}^{k+1}=x_i^k+{\nu }_i^k·\frac{d^k}{2},\\ \hfill x_{il}^{k+1}=x_i^k-{\nu }_i^k·\frac{d^k}{2},\end{array}\right.\end{array}$$

(17)

where $d^k$ is the distance between the left and right antenna, $d^k=\frac{{\delta }^k}{c}$, and $c=2$.

It is worth noting that there is a static nonlinear relationship between SOC and OCV. A fourth-order polynomial is adopted herein to characterize this nonlinear relationship,

$$\begin{array}{c}\hfill OCV=a_0+a_{1}SOC+a_{2}SO{C}^2+a_{3}SO{C}^3+a_{4}SO{C}^4,\end{array}$$

(18)

where $a_k$ ($k=0,1,\cdots ,4$) represent the polynomial coefficients.

The model parameters are identified in the time-domain and can be capsulated into a vector $\chi ={[a_o,a_1,a_2,a_3,a_4,R_o,R_1,C_1,R_2,C_2,W,{\alpha }_1,{\alpha }_2,{\alpha }_3]}^T$. The PI objective is to optimize the model parameters in order to minimize the error between the measured, $V_0\left(k\right)$, and the predicted, ${\widehat{V}}_o\left(k\right)$, voltages. Therefore, we have:

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

(19)

where $f(·)$ is the fitness function, and the constant M represents the number of sampling points.

To show the superiority of the novel HGA-FBSO algorithm, we list in

Table 1. The results of PI with the GA.

${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{R}}_0$	${\mathit{R}}_1$
$3.8594$	$1.0743$	$1.6648$	$1.5072$	$0.0959$	$0.0657$	1.3006
${\mathit{C}}_{\mathbf{1}}$	${\mathit{R}}_{\mathbf{2}}$	${\mathit{C}}_{\mathbf{2}}$	$\mathit{W}$	${\mathbf{\alpha }}_{\mathbf{1}}$	${\mathbf{\alpha }}_{\mathbf{2}}$	${\mathbf{\alpha }}_{\mathbf{3}}$
1361.59	0.8616	965.09	185.86	0.3180	0.8951	0.5305

,

Table 2. The results of PI with the FBSO.

${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{R}}_0$	${\mathit{R}}_1$
$3.8815$	$1.4725$	$2.8577$	$1.9817$	$-0.7167$	$0.0727$	1.0946
${\mathit{C}}_{\mathbf{1}}$	${\mathit{R}}_{\mathbf{2}}$	${\mathit{C}}_{\mathbf{2}}$	$\mathit{W}$	${\mathbf{\alpha }}_{\mathbf{1}}$	${\mathbf{\alpha }}_{\mathbf{2}}$	${\mathbf{\alpha }}_{\mathbf{3}}$
636.01	0.8419	778.31	548.29	0.4033	0.6411	0.5563

and

Table 3. The results of PI with the HGA-FBSO.

${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{R}}_0$	${\mathit{R}}_1$
$3.8643$	$1.1227$	$1.2584$	$-0.7111$	$-2.2234$	$0.0725$	0.9891
${\mathit{C}}_{\mathbf{1}}$	${\mathit{R}}_{\mathbf{2}}$	${\mathit{C}}_{\mathbf{2}}$	$\mathit{W}$	${\mathbf{\alpha }}_{\mathbf{1}}$	${\mathbf{\alpha }}_{\mathbf{2}}$	${\mathbf{\alpha }}_{\mathbf{3}}$
1101.56	0.4167	1892.57	1741.97	0.9477	0.4748	0.2180

the results yielded by the GA, FBSO and HGA-FBSO, respectively. The accuracy of the GA, FBSO and HGA-FBSO is shown in Figure 3, where the red, green and blue lines stand for GA, FBSO and HGA-FBSO, respectively. In addition, the measured values are represented by a black line. The absolute error (AE) and relative error (RE) of the three algorithms are shown in Figure 4 and Figure 5, respectively. To highlight the differences between the algorithms, Figure 4 and Figure 5 are partially magnified. One can verify that the maximum AE of the three methods are 0.2337 V, 0.1994 V and 0.1987 V, and the maximum RE are 7.27%, 6.27% and 6.25%, respectively. Consistent with our expectations, the proposed hybrid optimization algorithm can achieve more accurate results than the classical single algorithms.

Two additional metrics, namely, the RMSE and MAE are used to evaluate the algorithms accuracy. The RMSE and MAE are given by:

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

(20)

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

(21)

The RMSE and MAE of the three methods are summarized in

Table 4. The RMSE and MAE of the battery PI.

	GA	FBSO	HGA-FBSO
RMSE (mV)	15.5	11.4	10.5
MAE (%)	1.0786	0.7017	0.6058

. The HGA-FBSO is capable of predicting the battery voltage with a RMSE of 10.5 mV and a MAE of 0.6058%. This corresponds to accuracy improvements of 32.26% and 7.89% in terms of the RMSE, and of 43.83% and 13.67% in terms of the MAE, relative to the GA and FBSO, respectively. The results clearly show that the proposed hybrid method has higher accuracy than the traditional single optimization methods, and demonstrate the effectiveness of the new HGA-FBSO.

5. Conclusions

A new HGA-FBSO algorithm combining the advantages of the GA and BSO single optimization methods was proposed to improve the accuracy of PI of battery models. A FIM was adopted for the battery based on the EIS approach. The new HGA-FBSO introduced the concepts of crossover and mutation into the FBSO algorithm and updated the velocity formula based on FO concepts. The proposed scheme was shown to mitigate the issues revealed by single optimization methods, namely, poor stability and local optima trapping, while improving the optimization accuracy. The experimental results illustrated the effectiveness of the proposed method in the perspective of the RMSE and MAE of the voltage, with values of 10.5 mV and 0.6058%, which correspond to accuracy improvements of 32.26% and 7.89% for the RMSE, and 43.83% and 13.67% for the MAE, when compared with the GA and FBSO, respectively. The simulation results showed that the proposed method can significantly improve the PI quality of battery models. It should be pointed out that the change in step size and fractional order will affect the efficiency and effectiveness of the PI scheme. Therefore, in future work, we will address the impact of different parameter settings on the PI accuracy.