# Lightning Surge Analysis for Overhead Lines Considering Corona Effect
## 4 Transient Calculation Considering Corona Effect  
Figure 9a shows an overhead line with corona. It is subdivided into \(M\) line segments (see Figure 9b). On each line segment, the corona sheath is approximately considered to be uniform and replaced as a lumped non-linear branch carrying a corona current [24; 25], as shown in Figure 9c. After separating the corona sheath from each line segment, the remainder is free of corona and has the linear circuit parameters per unit length, i.e., \(R_{1}\), \(L_{1}\) and \(C_{1}\). Its equivalent circuit is depicted in Figure 10 [26], where \(\Delta R=\Delta Ik_{1}\), \(\Delta L=\Delta Ik_{1}\), \(\Delta C=\Delta\Gamma_{1}\), \(Z=(\Delta L/\Delta C)^{1/2}\) and \(\tau=\Delta l/v\) (\(v\) is the wave velocity). The expressions of the historical current sources \(I_{j-1}\) (\(t-\tau\)) and \(I_{j}\) (\(t-\tau\)) were also given in [26]. Considering Figures 9c and 10, the overhead line with corona is converted into a complete equivalent circuit, as shown in in Figure 11. This is a cascade circuit containing \(M\) non-linear branches carrying corona current. Topologically each unit including the non-linear branch is disconnected from other. The \(j\)th (\(j=1,2,\cdots\), \(M\)) circuit unit in Figure 11 can be represented as a one-port circuit, as shown in Figure 12. With the non-linear branch removed from the port (see Figure 13), the corresponding open-circuit voltage \(u_{OCj}\) and input impedance \(Z_{thj}\) is found by using the [PERSON]'s equivalent technique. The non-linear branch can be solved according to the [PERSON]'s equivalent circuit shown in Figure 14:  
\[u_{OCj}-Z_{thj}i_{cj}=u_{Cj}(j=1,2,\cdots,M) \tag{10}\]  
where the corona current \(i_{Cj}\) is described by Equations (1), (3) and (5). The fundamental studies made on an equivalent circuit model may refer to [27; 28; 29]. For the first curve section on the \(q\)-\(u\) curve, as described by Equation (1), if the corona charge satisfies \((C_{1}-C_{0})(u_{j}-U_{1})<q_{cj}<(C_{2}-C_{0})(u_{j}-U_{2})\), the corona current and branch voltage are given by:  
\[ \begin{array}{l}i_{Cj}=\frac{\beta}{\Gamma+\beta}[(C_{2}-C_{0})(u_{Cj}-U_{2 })-q_{CP}(t-\Delta t)]\\ u_{Cj}=\frac{\Gamma}{\Gamma+\Gamma}[u_{oc}+\Gamma U_{2}-\frac{\Gamma}{C_{2}-C _{0}}q_{CP}(t-\Delta t)]\end{array}  \tag{11}\]  
where \(\xi=\Delta t/2\) and \(\Gamma\) is,  
\[\Gamma=\frac{\beta Z_{thj}(C_{2}-C_{0})}{1+\beta\xi} \tag{12}\]  
Figure 9: Segmentation of an overhead line with corona. (**a**) An overhead line with corona, (**b**) line segment with corona, (**c**) non-linear branch carrying corona current. Figure 11: Complete equivalent circuit of an overhead line with corona. Figure 12: One-port circuit containing nonlinear branch. Figure 10: Equivalent circuit of a line segment free of corona. If the corona charge satisfies \(0<q_{cj}<(C_{1}-C_{0})(u_{j}-U_{1})\), the corona current and branch voltage are given by:  
\[ \begin{array}{l}i_{Cj}=\frac{1}{\Lambda_{1}}[\alpha(C_{1}-C_{0})(u_{Cj}-U_{1}) +\beta(C_{2}-C_{0})(u_{Cj}-U_{2})-(\alpha+\beta)q_{CP}(t-\Delta t)]\\ u_{Cj}=\frac{1}{\Lambda_{1}+\Lambda_{2}+\Lambda_{3}}[\Lambda_{1}u_{ocj}+ \Lambda_{2}U_{1}+\Lambda_{3}U_{2}+\Lambda_{4}q_{CP}(t-\Delta t)]\end{array}  \tag{13}\]  
where,  
\[ \begin{array}{l}\Lambda_{1}=1+\xi(\alpha+\beta)\\ \Lambda_{2}=Z_{thj}\alpha(C_{1}-C_{0})\\ \Lambda_{3}=Z_{thj}\beta(C_{2}-C_{0})\\ \Lambda_{4}=Z_{thj}(\alpha+\beta)\end{array}  \tag{14}\]  
For the second, third and subsequent curve segments on the \(q\)-\(u\) curve, the corresponding solutions of the corona current and branch voltage can be obtained in a manner similar to Equations (12) and (14). After determining \(i_{Cj}\) and \(u_{Cj}\) (\(j=1,2,\cdots,M\)), the nonlinear branch is replaced by a voltage source equal to \(u_{Cj}\). Figure 12 is converted into a purely linear circuit, as shown in Figure 15, and thereby the transient responses in the remaining linear part can be calculated by performing the nodal voltage analysis.