**3. New Strategy to Avoid the Impact on GC: Three-Point Climbing Algorithm**

The traditional sampling method when GC exists is shown in Figure 3. In the *M*<sup>1</sup> phase, *v*<sup>1</sup> decreases slowly, and *v*<sup>2</sup> increases slowly. Sampling continues for *v*<sup>1</sup> and *v*<sup>2</sup> until *t*11\_*<sup>n</sup>* and *t*12\_*n*, respectively; *v*<sup>1</sup> falls to stable value *vdcX*11, and *v*<sup>2</sup> rises to stable value *vdcX*12. After stabilization, the final *v*<sup>11</sup> and *v*<sup>12</sup> are obtained. The method switches to the *M*<sup>2</sup> phase until *v*<sup>1</sup> and *v*<sup>2</sup> reach stable values *vdcX*<sup>21</sup> and *vdcX*22, respectively, and the controller obtains the final *v*<sup>21</sup> and *v*22. One sampling period *T<sup>C</sup>* ends, and the insulation resistance values *Rf*<sup>2</sup> and *Rf*<sup>1</sup> are calculated by *v*<sup>11</sup> = *vdcX*11, *v*<sup>12</sup> = *vdcX*12, *v*<sup>21</sup> = *vdcX*21, and *v*<sup>22</sup> = *vdcX*22. The resistance values of *R*11, *R*12, *R*21, and *R*<sup>22</sup> are large, so the charging time of the capacitor is long. Sampling and calculation should be performed after GC charging is completed; hence, the measurement time of the traditional method is long and unable to meet the real-time requirements of EVs. To avoid the measurement overtime caused by GC, a new insulation resistance monitoring method, namely, three-point climbing algorithm, is proposed.

**Figure 3.** Traditional sampling method.

Figure 4 shows that each phase is sampled three times, and the sampling intervals are equal. With *v*<sup>11</sup> as an example, the three sampling times are *t*11\_1, *t*11\_2, and *t*11\_3; the sampling voltage values are *v*11\_1, *v*11 \_2, and *v*11\_3, respectively, and the time intervals are ∆*t*. Similar definitions of *v*12, *v*21, and *v*<sup>22</sup> are provided to facilitate the calculation, and *E*<sup>1</sup> and *E*<sup>2</sup> are used to present natural exponential function. The following equation is then obtained from (5).

$$\begin{cases} \frac{v\_{11\perp} - v\_{11\perp}}{v\_{11\perp} - v\_{11\perp}} = e^{-\frac{\Delta t}{v\_1}} = E\_1, \quad \frac{v\_{12\perp} - v\_{12\perp}}{v\_{12\perp} - v\_{12\perp}} = e^{-\frac{\Delta t}{v\_1}} = E\_1\\\\ \frac{v\_{21\perp} - v\_{21\perp}}{v\_{21\perp} - v\_{21\perp}} = e^{-\frac{\Delta t}{v\_2}} = E\_2, \quad \frac{v\_{22\perp} - v\_{22\perp}}{v\_{22\perp} - v\_{22\perp}} = e^{-\frac{\Delta t}{v\_2}} = E\_2 \end{cases} \tag{6}$$

*t*11\_1, *t*12\_1, *t*21\_1, and *t*22\_1 are set as initial times for the first-order circuit full response curve, and the curve Equation (5) is converted into Equation (7).

$$\begin{cases} v\_{11\\_2} = v\_{dc}X\_{11} + (v\_{11\\_1} - v\_{dc}X\_{11})E\_1\\ v\_{12\\_2} = v\_{dc}X\_{12} + (v\_{12\\_1} - v\_{dc}X\_{12})E\_1\\ v\_{21\\_2} = v\_{dc}X\_{21} + (v\_{21\\_1} - v\_{dc}X\_{21})E\_2\\ v\_{22\\_2} = v\_{dc}X\_{22} + (v\_{22\\_1} - v\_{dc}X\_{22})E\_2 \end{cases} \tag{7}$$

Equation (7) is converted into Equation (8).

$$\begin{cases} X\_{11} = \frac{v\_{11,2} - v\_{11,1}E\_1}{v\_{dc}(1 - E\_1)}, & X\_{12} = \frac{v\_{12,2} - v\_{12,1}E\_1}{v\_{dc}(1 - E\_1)} \\\\ X\_{21} = \frac{v\_{21,2} - v\_{12,1}E\_2}{v\_{dc}(1 - E\_2)}, & X\_{22} = \frac{v\_{22,2} - v\_{22,1}E\_2}{v\_{dc}(1 - E\_2)} \end{cases} \tag{8}$$

*X*11, *X*12, *X*21, and *X*<sup>22</sup> in Equation (8) are known values calculated by sampling voltage. Equation (9) can be obtained from Equations (2), (3), and (8), and insulation resistance values *Rf*<sup>1</sup> and *Rf*<sup>2</sup> can be solved.

$$\begin{cases} R\_{f1} = \frac{R\_a R\_b}{\frac{R\_a - R\_b}{X\_{12} - X\_{22}} X\_{12} - R\_a}, R\_{f2} = \frac{R\_a R\_b}{\frac{R\_b - R\_a}{X\_{11} - X\_{21}} X\_{11} - R\_b} \\ R\_{f1} = \frac{R\_a R\_b}{\frac{R\_a - R\_b}{X\_{12} - X\_{22}} X\_{22} - R\_b}, R\_{f2} = \frac{R\_a R\_b}{\frac{R\_b - R\_a}{X\_{11} - X\_{21}} X\_{21} - R\_a} \end{cases} \tag{9}$$

**Figure 4.** Proposed three-point climbing algorithm.
