B. Appendix

Based on the axisymmetric character of sample configuration in our model, the cylindrical coordinate system is employed, so the convection term could be rewritten as

$$\frac{\partial w}{\partial t} = -\frac{\partial f}{\partial r} - \frac{\partial g}{\partial z} \tag{A10}$$

where w = rN, f = rNWr, g = rNWz, W = Wrer + Wzez, er, and ez are the unit vectors along r and z directions, respectively. To solve this equation, six steps are needed:

(1) to obtain the low-order flux FL iþ1 2,j , G<sup>L</sup> i,jþ<sup>1</sup> 2

A. Appendix

field in V/cm

W<sup>e</sup> ¼

8 >>>><

>>>>:

α

(

<sup>N</sup> <sup>¼</sup> <sup>8</sup>:<sup>889</sup> � <sup>10</sup>�<sup>5</sup>

η2

B. Appendix

The transportation parameters for air are expressed by the following equations:

<sup>N</sup> = 2.69 � 1019 cm�<sup>3</sup> indicates the number of gas molecules per unit volume, and <sup>E</sup> is the local

�ð Þ <sup>E</sup>=j j <sup>E</sup> <sup>1</sup>:<sup>03</sup> � 1022j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>þ</sup> <sup>1</sup>:<sup>3</sup> � 106 � � <sup>10</sup>�<sup>16</sup> <sup>≤</sup> j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>≤</sup> <sup>2</sup>:<sup>0</sup> � <sup>10</sup>�<sup>15</sup> �ð Þ <sup>E</sup>=j j <sup>E</sup> <sup>7</sup>:<sup>2973</sup> � 1021j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>þ</sup> <sup>1</sup>:<sup>63</sup> � 106 � � <sup>2</sup>:<sup>6</sup> � <sup>10</sup>�<sup>17</sup> <sup>≤</sup> j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>&</sup>lt; <sup>10</sup>�<sup>16</sup>

<sup>W</sup><sup>n</sup> <sup>¼</sup> �2:7E Ej j=<sup>N</sup> <sup>&</sup>gt; <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>16</sup>

�1:86E Ej j=<sup>N</sup> <sup>≤</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>16</sup>

j j <sup>E</sup> <sup>=</sup><sup>N</sup> j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>&</sup>gt; <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>15</sup>

j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>þ</sup> <sup>2</sup>:<sup>567</sup> � <sup>10</sup>�<sup>19</sup> j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>&</sup>gt; <sup>1</sup>:<sup>05</sup> � <sup>10</sup>�<sup>15</sup>

j j <sup>E</sup> <sup>=</sup><sup>N</sup> � <sup>2</sup>:<sup>893</sup> � <sup>10</sup>�<sup>19</sup> j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>≤</sup> <sup>1</sup>:<sup>05</sup> � <sup>10</sup>�<sup>15</sup>

j j <sup>E</sup> <sup>=</sup><sup>N</sup> j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>≤</sup> <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>15</sup>

<sup>η</sup>3=N<sup>2</sup> <sup>¼</sup> <sup>4</sup>:<sup>7778</sup> � <sup>10</sup>�<sup>59</sup>ð Þ j j <sup>E</sup> <sup>=</sup><sup>N</sup> �1:<sup>2749</sup> (A6)

W<sup>p</sup> ¼ 2:34E (A3)

η ¼ η<sup>2</sup> þ η<sup>3</sup> (A7)

<sup>β</sup> <sup>¼</sup> <sup>2</sup>:<sup>0</sup> � <sup>10</sup>�<sup>7</sup> (A8)

ð Þ j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>0</sup>:<sup>54069</sup>j j We=<sup>E</sup> (A9)

<sup>∂</sup><sup>z</sup> (A10)

(A1)

(A2)

(A4)

(A5)

�ð Þ <sup>E</sup>=j j <sup>E</sup> <sup>7</sup>:<sup>4</sup> � <sup>10</sup><sup>21</sup>j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>þ</sup> <sup>7</sup>:<sup>1</sup> � <sup>10</sup><sup>6</sup> � � j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>&</sup>gt; <sup>2</sup>:<sup>0</sup> � <sup>10</sup>�<sup>15</sup>

�ð Þ <sup>E</sup>=j j <sup>E</sup> <sup>6</sup>:<sup>87</sup> � 1022j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>þ</sup> <sup>3</sup>:<sup>38</sup> � 104 � � j j <sup>E</sup> <sup>=</sup><sup>N</sup> <sup>&</sup>lt; <sup>2</sup>:<sup>6</sup> � <sup>10</sup>�<sup>17</sup>

�7:248�10�<sup>15</sup>

where η<sup>2</sup> and η<sup>3</sup> are the two-body and three-body attachment coefficients, respectively.

Based on the axisymmetric character of sample configuration in our model, the cylindrical

where w = rN, f = rNWr, g = rNWz, W = Wrer + Wzez, er, and ez are the unit vectors along r and z

<sup>D</sup> <sup>¼</sup> <sup>0</sup>:<sup>3341</sup> � 109

coordinate system is employed, so the convection term could be rewritten as

directions, respectively. To solve this equation, six steps are needed:

∂w <sup>∂</sup><sup>t</sup> ¼ � <sup>∂</sup><sup>f</sup> ∂r � ∂g

�5:593�10�<sup>15</sup>

(

<sup>6</sup>:<sup>619</sup> � <sup>10</sup>�<sup>17</sup><sup>e</sup>

<sup>N</sup> <sup>¼</sup> <sup>2</sup>:<sup>0</sup> � <sup>10</sup>�<sup>16</sup><sup>e</sup>

8 < :

124 Plasma Science and Technology - Basic Fundamentals and Modern Applications

<sup>6</sup>:<sup>089</sup> � <sup>10</sup>�<sup>4</sup>

$$\mathbf{F}^{\mathbb{L}}\_{i+\frac{1}{2}j} = (2i+1)\pi\Delta\mathrm{z}\Delta\mathrm{r}\Delta\mathrm{t}(\mathcal{W}\_{r})\_{i+\frac{1}{2}j}\mathfrak{w}\begin{cases} \boldsymbol{w} = \mathrm{i}\boldsymbol{\Lambda}\boldsymbol{r}\mathrm{N}\_{i,j} \quad (\mathcal{W}\_{r})\_{i+\frac{1}{2}j} \ge 0\\ \boldsymbol{w} = (i+1)\boldsymbol{\Lambda}\boldsymbol{r}\mathrm{N}\_{i+1,j} \left(\mathcal{W}\_{r}\right)\_{i+\frac{1}{2}j} < 0 \end{cases} \tag{A11}$$

$$\mathbf{G}^{L}\_{\ i,j+\frac{1}{2}} = \Lambda t \mathbf{S}\_{i} (\mathbf{W}\_{z})\_{i,j+\frac{1}{2}} w \begin{cases} w = \mathrm{i}\Lambda r \mathrm{N}\_{i,j} \ (\mathbf{W}\_{z})\_{i,j+\frac{1}{2}} \ge 0 \\ w = \mathrm{i}\Lambda r \mathrm{N}\_{i,j+1} \ (\mathbf{W}\_{z})\_{i,j+\frac{1}{2}} < 0 \end{cases} \tag{A12}$$

$$S\_i = 2i\pi\Delta r^2\tag{A13}$$

$$(\mathcal{W}\_r)\_{i+\frac{1}{2}j} = \frac{(\mathcal{W}\_r)\_{i,j} + (\mathcal{W}\_r)\_{i+1,j}}{2} \tag{A14}$$

$$(\mathcal{W}\_z)\_{i,j+\frac{1}{2}} = \frac{(\mathcal{W}\_z)\_{i,j} + (\mathcal{W}\_z)\_{i,j+1}}{2} \tag{A15}$$

where i and j are the sequence number of node along r and z directions, respectively. (2) to obtain high-order flux F<sup>H</sup> iþ1 2,j , G<sup>H</sup> i,jþ<sup>1</sup> 2

$$\begin{split} F^{H}\_{i+\frac{1}{2}j} &= (2i+1)\pi\Delta z\Delta r\Delta t \left[ \frac{533}{840} \left( f\_{i+1,j} + f\_{i,j} \right) - \frac{139}{840} \left( f\_{i+2,j} + f\_{i-1,j} \right) \right. \\ &\left. + \frac{29}{840} \left( f\_{i+3,j} + f\_{i-2,j} \right) - \frac{1}{280} \left( f\_{i+4,j} + f\_{i-3,j} \right) \right] \end{split} \tag{A16}$$

$$\begin{split} G^{H}\_{i,j+\frac{1}{2}} &= \mathbb{S}\_{i} \Delta t \left[ \frac{533}{840} \left( \mathbf{g}\_{i,j+1} + \mathbf{g}\_{i,j} \right) - \frac{139}{840} \left( \mathbf{g}\_{i,j+2} + \mathbf{g}\_{i,j-1} \right) + \frac{29}{840} \left( \mathbf{g}\_{i,j+3} + \mathbf{g}\_{i,j-2} \right) \right. \\ &\left. - \frac{1}{280} \left( \mathbf{g}\_{i,j+4} + \mathbf{g}\_{i,j-3} \right) \right] \end{split} \tag{A17}$$

$$f\_{i,j} = \mathrm{i}\Delta r \mathrm{N}\_{i,j}(\mathcal{W}\_r)\_{i,j} \tag{A18}$$

$$\mathbf{g}\_{i,j} = \mathbf{i} \Delta r \mathbf{N}\_{i,j} (\mathbf{W}\_z)\_{i,j} \tag{A19}$$

(3) to define antidiffusion flux

$$A\_{i+\frac{1}{2}j} \equiv F^{H\_{i+\frac{1}{2}j}} - F^{L}\_{i+\frac{1}{2}j} \tag{A20}$$

$$\mathbf{A}\_{i,j+\frac{1}{2}} \equiv \mathbf{G}^{H}\_{\ i,j+\frac{1}{2}} - \mathbf{G}^{L}\_{\ i,j+\frac{1}{2}} \tag{A21}$$

(4) to obtain the temporary solution

$$
\Delta w\_{i,j}^{\text{td}} = w\_{i,j}^n - \Delta V\_{i,j}^{-1} \left[ F\_{i+\frac{1}{2}j}^L - F\_{i-\frac{1}{2}j}^L + G\_{i,j+\frac{1}{2}}^L - G\_{i,j-\frac{1}{2}}^L \right] \tag{A22}
$$

$$w\_{i,j} = \mathrm{i}\Delta r \mathrm{N}\_{i,j} \tag{A23}$$

$$
\Delta V\_{i,j} = \Delta \mathbf{z} \mathbf{S}\_i \tag{A24}
$$

R�

(6) to solve the charge concentration

Cheng Pan\*, Ju Tang and Fuping Zeng

Author details

References

190

44(24):21

wnþ<sup>1</sup> i,j <sup>¼</sup> wtd

where n indicates nΔt and n+1 indicates ð Þ n þ 1 Δt.

\*Address all correspondence to: pancheng2036@gmail.com School of Electrical Engineering, Wuhan University, China

Electrical Insulation. 2005;12(5):905-913

Electrical Insulation. 1991;26(5):902-948

Heinemann; 1989

i,j <sup>¼</sup> min 1; <sup>Q</sup>�

i,j AC iþ1 <sup>2</sup>,j � AC i�1 <sup>2</sup>,j <sup>þ</sup> AC i,jþ<sup>1</sup> 2 � <sup>A</sup><sup>C</sup> i,j�<sup>1</sup> 2 h i (A41)

8 < :

i,j � <sup>Δ</sup>V�<sup>1</sup>

i,j =P� i,j � � <sup>P</sup>�

0 P�

[1] Morshuis PHF. Degradation of solid dielectrics due to internal partial discharge: Some thoughts on progress made and where to go now. IEEE Transactions on Dielectrics and

[2] Van Brunt RJ. Stochastic properties of partial-discharge phenomena. IEEE Transactions on

[3] Kreuger FH. Partial Discharge Detection in High-Voltage Equipment. Oxford: Butterworth-

[4] Gutfleisch F, Niemeyer L. Measurement and simulation of PD in epoxy voids. IEEE Trans-

[5] Das P, Chakravorti S. Simulation of PD patterns due to a narrow void in different E-field

[6] Wu K, Suzuoki Y, Mizutani T, Xie H. A novel physical model for partial discharge in narrow channels. IEEE Transactions on Dielectrics and Electrical Insulation. 1999;6(2):181-

[7] Illias HA, Chen G, Lewin PL. The influence of spherical cavity surface charge distribution on the sequence of partial discharge events. Journal of Physics D: Applied Physics. 2011;

[8] Meek JM, Craggs JD. Electrical Breakdown of Gases. New York: Wiley; 1978

actions on Dielectrics and Electrical Insulation. 1995;2(5):729-743

distribution. Journal of Electrostatics. 2010;68(3):218-226

i,j > 0

Numerical Modeling of Partial Discharge Development Process

http://dx.doi.org/10.5772/intechopen.79215

(A40)

127

i,j ¼ 0

(5) to restrict the antidiffusion flux

$$\text{If } A\_{i + \frac{1}{2}j} \left( w\_{i+1,j}^{td} - w\_{i,j}^{td} \right) < 0 \\ \text{6\n\nA} \left\{ A\_{i + \frac{1}{2}j} \left( w\_{i+2,j}^{td} - w\_{i+1,j}^{td} \right) < 0 \\ \left| A\_{i + \frac{1}{2}j} \left( w\_{i,j}^{td} - w\_{i-1,j}^{td} \right) < 0 \right. \right. \\ \left. \left. \left. \left( \begin{array}{c} -\frac{1}{2} \\ \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) \right) - \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) \right. \\ \left. \left. \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) - \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) \right) \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right. \\ \left. \left. \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) - \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) \right) \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right. \\ \left. \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) - \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right) \right) \left( \begin{array}{c} -\frac{1}{2} \\ \end{array} \right. \\ \left. \end{$$

$$\text{If } A\_{i, j + \frac{1}{2}} \left( w\_{i, j + 1}^{ld} - w\_{i, j}^{td} \right) < 0 \& \left\{ A\_{i, j + \frac{1}{2}} \left( w\_{i, j + 2}^{ld} - w\_{i, j + 1}^{td} \right) < 0 \right\} \\ A\_{i, j + \frac{1}{2}} \left( w\_{i, j}^{td} - w\_{i, j - 1}^{td} \right) < 0 \\ \text{;} \\ A\_{i, j + \frac{1}{2}} = 0 \end{aligned} \tag{A26}$$

$$A\_{i+\frac{1}{2}j}^{\mathbb{C}} = \mathbb{C}\_{i+\frac{1}{2}j} A\_{i+\frac{1}{2}j} \quad 0 \le \mathbb{C}\_{i+\frac{1}{2}j} \le 1 \tag{A27}$$

$$A\_{i,j+\frac{1}{2}}^{\mathbb{C}} = \mathbb{C}\_{i,j+\frac{1}{2}} A\_{i,j+\frac{1}{2}} \quad 0 \le \mathbb{C}\_{i,j+\frac{1}{2}} \le 1 \tag{A28}$$

$$\mathbf{C}\_{i+\frac{1}{2}j} = \begin{cases} \min\left(\mathbf{R}^+\_{i+1,j}, \mathbf{R}^-\_{i,j}\right) & A\_{i+\frac{1}{2}j} \ge 0 \\\min\left(\mathbf{R}^+\_{i,j}, \mathbf{R}^-\_{i+1,j}\right) & A\_{i+\frac{1}{2}j} < 0 \end{cases} \tag{A29}$$

$$\mathbf{C}\_{i,j+\frac{1}{2}} = \begin{cases} \min\left(\mathbf{R}\_{i,j+1}^{+}, \mathbf{R}\_{i,j}^{-}\right) & A\_{i,j+\frac{1}{2}} \ge 0 \\\\ \min\left(\mathbf{R}\_{i,j}^{+}, \mathbf{R}\_{i,j+1}^{-}\right) & A\_{i,j+\frac{1}{2}} < 0 \end{cases} \tag{A30}$$

$$w\_{i,j}^{a} = \max\left(w\_{i,j}^{n}, w\_{i,j}^{td}\right) \tag{A31}$$

$$\mathfrak{w}\_{i,j}^{\max} = \max \left( \mathfrak{w}\_{i-1,j}^{a}, \mathfrak{w}\_{i,j}^{a}, \mathfrak{w}\_{i+1,j}^{a}, \mathfrak{w}\_{i,j-1}^{a}, \mathfrak{w}\_{i,j+1}^{a} \right) \tag{A32}$$

$$w\_{i,j}^b = \min\left(w\_{i,j}^n, w\_{i,j}^{td}\right) \tag{A33}$$

$$w\_{i,j}^{\min} = \min\left(w\_{i-1,j}^b, w\_{i,j}^b, w\_{i+1,j}^b, w\_{i,j-1}^b, w\_{i,j+1}^b\right) \tag{A34}$$

$$P\_{i,j}^+ = \max\left(0, A\_{i - \frac{1}{2}j}\right) - \min\left(0, A\_{i + \frac{1}{2}j}\right) + \max\left(0, A\_{i, j - \frac{1}{2}}\right) - \min\left(0, A\_{i, j + \frac{1}{2}}\right) \tag{A35}$$

$$Q^+\_{i,j} = \left(w^{\text{max}}\_{i,j} - w^{td}\_{i,j}\right) \Delta V\_{i,j} \tag{A36}$$

$$\mathcal{R}\_{i,j}^+ = \begin{cases} \min\left(1, \mathbb{Q}\_{i,j}^+/P\_{i,j}^+\right) & P\_{i,j}^+ > 0\\ 0 & P\_{i,j}^+ = 0 \end{cases} \tag{A37}$$

$$P\_{i,j}^- = \max\left(0, A\_{i + \frac{1}{2}j}\right) - \min\left(0, A\_{i - \frac{1}{2}j}\right) + \max\left(0, A\_{i, j + \frac{1}{2}}\right) - \min\left(0, A\_{i, j - \frac{1}{2}}\right) \tag{A38}$$

$$Q\_{i,j}^- = \left(w\_{i,j}^{td} - w\_{i,j}^{\min}\right) \Delta V\_{i,j} \tag{A39}$$

Numerical Modeling of Partial Discharge Development Process http://dx.doi.org/10.5772/intechopen.79215 127

$$R\_{i,j}^- = \begin{cases} \min\left(1, Q\_{i,j}^- / P\_{i,j}^-\right) & P\_{i,j}^- > 0\\ 0 & P\_{i,j}^- = 0 \end{cases} \tag{A40}$$

(6) to solve the charge concentration

$$w\_{i,j}^{n+1} = w\_{i,j}^{td} - \Delta V\_{i,j}^{-1} \left[ A\_{i + \frac{1}{2}j}^{\mathbb{C}} - A\_{i - \frac{1}{2}j}^{\mathbb{C}} + A\_{i, j + \frac{1}{2}}^{\mathbb{C}} - A\_{i, j - \frac{1}{2}}^{\mathbb{C}} \right] \tag{A41}$$

where n indicates nΔt and n+1 indicates ð Þ n þ 1 Δt.
