*2.2. Establishment of Lithium-Ion Battery FOM*

The transfer function is presented in Equation (9) for the lithium-ion battery FOM as follows:

$$\frac{\mathcal{U}\_{\rm CCV}(\mathbf{s}) - \mathcal{U}\_{\rm d}(\mathbf{s})}{I(\mathbf{s})} = \frac{R\_1}{1 + R\_1 Z\_1^{-1}} + \frac{R\_2}{1 + R\_2 Z\_2^{-1}} + Z\_W + R\_0 \tag{9}$$

where *Z*<sup>1</sup> = (*CPE*1*S*α) <sup>−</sup><sup>1</sup> and *<sup>Z</sup>*<sup>2</sup> <sup>=</sup> *CPE*1*S*<sup>β</sup> <sup>−</sup><sup>1</sup> indicate the impedances of *CPE*<sup>1</sup> and *CPE*<sup>2</sup> element, respectively. The impedance of the Wahlberg element is denoted by *ZW* = (*WS*γ) −1 .

Assuming the system input is *u* = *I(t)*, and the output is *y* = *UOCV (t)* − *Ud(t)*, then the system model is denoted by a fractional calculus equation, as shown in Equation (10).

(*WD*<sup>γ</sup> + *WR*1*CPE*1*D*α+<sup>γ</sup> + *WR*2*CPE*2*D*β+<sup>γ</sup> + *WR*1*CPE*1*R*2*CPE*2*D*α+β+γ)*y*(*t*) = [*R*1*CPE*1*D*<sup>α</sup> + *R*2*CPE*2*D*<sup>β</sup> + (*R*<sup>0</sup> + *R*<sup>1</sup> + *R*2)*WD*γ+ *R*1*CPE*1*R*2*CPE*2*D*α+<sup>β</sup> + (*R*<sup>0</sup> + *R*2)*WR*1*CPE*1*D*α+γ+ (*R*<sup>0</sup> + *R*1)*WR*2*CPE*2*D*β+<sup>γ</sup> + *R*0*WR*1*CPE*1*R*2*CPE*2*D*α+β+γ]*u*(*t*) + *u*(*t*) (10)

Here, the fractional-orders of *CPE*<sup>1</sup> and *CPE*<sup>2</sup> are represented by α and β, and the fractional-order of the Wahlberg element is denoted by γ. The parameters *D*γ, *D*α, *D*β, *D*α+γ, *D*β+γ, *D*α+β, *D*α+β+<sup>γ</sup> indicate the fractional-order operators.

The definition of SOC of the FOM is given by:

$$D^1 SOC(t) = -\frac{\eta}{\mathbb{C}\_{\mathbf{n}}} \mu(t) \tag{11}$$

where *C*<sup>n</sup> is the rated capacity of the battery; η stands for the battery coulomb efficiency.

By using calculations as presented in [13], we can transform Equation (10) into a first-order difference equation as follows:

$$y(k) = -\frac{A(1)}{A(0)}y(k-1) + \frac{1+B(0)}{A(0)}u(k) + \frac{B(1)}{A(0)}u(k-1)\tag{12}$$

Discretizing Equation (11):

$$\text{SOC}(k) = \text{SOC}(k-1) - \frac{\eta T}{\mathbb{C}\_n} \mu(k) \tag{13}$$

The FOMs of lithium-ion batteries can be established using Equations (12) and (13), and the parameters that the model needs to identify are as follows:

$$
\theta = \begin{bmatrix} R\_0 & R\_1 \ \text{CPE}\_1 \ R\_2 \ \text{CPE}\_2 \ W \ \alpha \ \beta \ \gamma \end{bmatrix} \tag{14}
$$
