*2.1. Establishment of Lithium-Ion Battery DPM*

The terminal voltage of the battery is shown as follows:

$$
\mathcal{U}L\_d = \mathcal{U}\_{\rm CCV} + \mathcal{R}\_0 I + \mathcal{U}\_1 + \mathcal{U}\_2 \tag{1}
$$

The changes in the rates of voltages *<sup>U</sup>*<sup>1</sup> and *<sup>U</sup>*<sup>2</sup> are • *<sup>U</sup>*<sup>1</sup> and • *U*2, respectively, which can be expressed by Equations (2) and (3):

$$
\hat{\mathbf{U}}\_1 = -\frac{\mathbf{U}\_1}{R\_1 \mathbf{C}\_1} + \frac{\mathbf{I}}{\mathbf{C}\_1} \tag{2}
$$

$$
\hat{\mathbf{U}}\_2 = -\frac{\mathbf{U}\_2}{R\_2 \mathbf{C}\_2} + \frac{\mathbf{I}}{\mathbf{C}\_2} \tag{3}
$$

The definition of SOC is presented in Equations (4) as follows:

$$\text{SOC}(t) = \text{SOC}(t\_0) + \int\_{t\_0}^{t} \frac{\eta I(\tau)}{Q} d\tau \tag{4}$$

where the values of SOC at time *t* and *t0* are denoted by *SOC(t)* and *SOC(t0)*, respectively, *Q* stands for the maximum available capacity, the charging and discharging efficiency is represented by η. Supposing Δ*T* indicates the sampling time, discretizing Equations (1), (2), and (4) as follows:

$$\mathcal{U}\_{1}(k) = \exp(\frac{-\Delta T}{R\_{1}\mathcal{C}\_{1}})lI\_{1}(k-1) + R\_{1}l(k)[1 - \exp(\frac{-\Delta T}{R\_{1}\mathcal{C}\_{1}})] \tag{5}$$

$$\mathcal{U}l\_2(k) = \exp(\frac{-\Delta T}{R\_2 C\_2})\mathcal{U}\_2(k-1) + R\_2 I(k)[1 - \exp(\frac{-\Delta T}{R\_2 C\_2})] \tag{6}$$

$$SOC(k) = SOC(k-1) + \frac{\eta \Delta T}{Q} I(k) \tag{7}$$

The parameters to be identified are:

$$\theta = \begin{bmatrix} \mathcal{R}\_0 \ R\_1 \ \mathcal{C}\_1 \ R\_2 \ \mathcal{C}\_2 \ \mathcal{U}\_{\text{OCV}} \ \mathcal{Q} \end{bmatrix} \tag{8}$$
