*3.3. Mathematical Expression*

For better application in EMS, we further establish the mathematical expression of the model parameters. The method contains the following two steps.


$$\begin{cases} \mathcal{U}\_{\rm loc}(z) = a\_1 z^3 + a\_2 z^2 + a\_3 z + a\_4 + a\_5 \exp(-\frac{a\_6}{z})\\\mathcal{R}\_0(z) = b\_1 z^4 + b\_2 z^3 + b\_3 z^2 + b\_4 z + b\_5 \\\overline{\mathcal{R}}\_p = \frac{\sum\_p \mathcal{R}\_p^{(j)}}{n} \\\ \overline{\tau} = \frac{\sum z^{(j)}}{n} \end{cases} \tag{20}$$

where *a*1–*a*<sup>6</sup> and *b*1–*b*<sup>5</sup> are coefficients, *n* is the total number of segments.

**Figure 4.** Identification results on data segments of four cells: (**A**) open circuit voltage (OCV); (**B**) internal resistance; (**C**) time constant; (**D**) resistance of RC network.

Based on the above expressions, GA is carried out once again to perform the identification of parameters on the entire SOC range under each aging condition. The coefficients for GA to optimize are rewritten as

$$\rho(SOH) = [a\_1, a\_2, \dots, a\_6, b\_1, b\_2, \dots, b\_5, R\_p, \overline{\tau}] \tag{21}$$

in which *a*1,..., *a*6, *b*1,..., *b*<sup>5</sup> are subject to

$$\begin{cases} a\_i \in [a\_{i,\text{min}}, a\_{i,\text{max}}]\_\prime \text{ } i = 1, 2, \dots, 6\\\ b\_j \in [b\_{j,\text{min}}, b\_{j,\text{max}}]\_\prime \text{ } j = 1, 2, \dots, 5 \end{cases}$$

where *ai*,min, *ai*,max, *bj*,min, and *bj*,max are limitations of the coefficients' boundary.
