## 1 Introduction  
Lightning is one of the most important issues in terms of protection of transmission and distribution lines [1]. Their protection requires an accurate evaluation of the insulation coordination system [2] as well as a correct computation of the number of dangerous events striking the line per year. The latter parameter is usually computed through the lightning performance procedure--a high number of lightning events, each of them characterized by different parameters extracted from their own Probability Density Function (PDF), is generated and their effects on the power system are computed through a simplified method [1; 3; 4] or through a numerical code [5; 6; 7; 8; 9]. The number of events that exceeds a threshold value depending on the line Critical Flashover value (CFO) is considered dangerous. Each event can be classified as a direct or indirect stroke in accordance to the electrogeometrical criterion [10]. When we deal with transmission lines, characterized by high CFO, the majority of the dangerous events is represented by direct strokes, while dealing with overhead distribution lines, indirect strokes are the most affecting category. The lightning performance procedure of overhead distribution lines has been deeply studied and optimized by several researchers and usually requires the use of a numerical code. Among them, Reference [7] proposes a procedure based on LIOV code which takes into account a triangular waveform for the channel-base current and the possibility to consider complex power systems as well as a finite ground conductivity. The authors of Reference [11] provide an application of the recursive stratified sampling technique in order to reduce the computational effort. In Reference [6], the authors propose a procedure which can be extended to whatever channel-base current based on the construction of an electromagnetic field database. The corona effect, that is, the process that describes an electrical discharge by the ionization of the fluid that surrounds a conductor, usually, is not considered in the lightning performance of distribution lines (i.e., taking into account both direct and indirect events): typically due to its occurrence only for small conductor diameters and in case of direct events, which represents the less significant part. However, as pointed out in this work, there are some cases where the corona discharge occurs also for indirect events, leading to a meaningful impact in the lightning performance evaluation. When one deals with direct events, the corona discharge affects the surge propagation more than the ground resistivity but reduces the voltages induced on the line [12]. On the other side, when one deals with indirect strokes, Reference [12], _\"computation results showed a significant increase in the amplitude of the induced voltages in presence of corona\"_. The result has been confirmed by Reference [13]. The implementation of corona discharge in the computation of lightning-induced voltages has been based on two different strategies: (i) The concept of dynamic capacitance has been proposed in Reference [12] that is, when the induced voltage overcomes a threshold, the corona discharge determines an increase of the per unit length capacitance involved in the Agrawal model [14]; consequently, the dynamic capacitance mimics the experimental \(q\)-\(v\) characteristic of the corona discharge [15]. (ii) In References [16,17] the 3D-FDTD code simulates the corona discharge giving different values to the conductivity of the cell where corona effect is located; the main difference with the previous approach is that here the \(q\)-\(v\) characteristic results as an output calculation from the numerical integrations, while in the previous one appears as an input of the problem. However, both approaches lead to the same conclusions, that is, the enhancement of the induced voltage.