**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 [5–9]. 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. Unfortunately, to the best of author's knowledge, both approaches limit their studies to the evaluation of the effect of a single event striking in the proximity of the line and a parametric analysis aimed at relating the most affecting parameters (line diameter, environmental conditions) to the induced voltage is missing.

The aim of this work is to evaluate the effects of corona discharge in the lightning performance computation of an overhead distribution line and to make some sensitivity analysis on the parameters that mainly affect the enhancement of the number of dangerous events striking the line. The implementation of corona discharge will be based on the procedure described in Reference [5] and validated in Reference [18,19] .

The paper is structured as follows: Section 2 recalls the concepts of the corona discharge, Section 3 describes the implementation of corona discharge in the procedure developed in Reference [5] and Section 4 focuses on the lightning performance computation. Later, the sensitivity analysis on the main parameters (line diameter and environmental conditions) affecting the enhancement of dangerous events due to corona is proposed in Section 5. Finally, in Section 6, some conclusions are drawn.

### **2. The Corona Effect**

According to Reference [20], the corona discharge can occur when the electric field in air in the vicinity of object at high voltages or exposed to high electric fields may overhelm the critical electric field able to create electron avalanches in air. The corona discharge can be either a positive of negative discharge and, according to Reference [21], the corona occurs on a conductor when the electric field on its surface is higher than the critical field *Ec*.

$$E\_c = m \times 2.594 \times 10^6 \left( 1 + \frac{0.1269}{r^{0.4346}} \right),\tag{1}$$

where *m* is a surface state coefficient quantifying the irregularities of the cable and generally deduced from tests. This formula has been developed assuming 20 ◦C, a pressure of 760 mmHg and a humidity of the air equal to 11 g/m3. Please note that *r* is the radius of the conductor and is expressed in cm. It is important to notice that according to Reference [21] the variation of humidity changes the critical field *Ec*. With respect to (1) an increase of the air humidity to 18 g/m<sup>3</sup> leads to an increase of the critical

electric field of 2%. Figure 4 of Reference [21] showed the variation of the critical electric field as a function of the air humidity.

In Reference [22], recently, the authors provided a new expression for the critical electric field under variable atmospheric conditions, which is here proposed for sake of completeness.

$$E\_{\rm c} = 31.53 \left( 1 + \frac{A}{K^a n\_s^b r^c} \right),\tag{2}$$

where

$$K = \delta^{1.01} \left( 1 + 0.08 \left( \frac{H}{11}^{0.72} - 1 \right) \right),\tag{3}$$

being *δ* the relative air density, *H* the air humidity, *ns* a coefficient depending on the number of strands in the outer layer of the conductor and *A*, *a*, *b*, *c* coefficients depending on the voltage polarity (Table 1 of Reference [22]).

In addition to this, the corona discharge occurs if a free electron is available at the instant when the electric field overcomes the critical value in (1). It means that there is a certain time lag between the application of the electric field and the time of creation of a free electron in the gas volume [20]: this time is known as the *statistical time lag* or *inception time delay*. The statistical time lag decreases when the applied electric field increases .

Once a free electron is found, the corona discharge occurs, but its sustainment is achieved only if the electric field in front of the streamers is not lower than 4 to 5 kV/cm (positive streamers) and 11 to 18 kV/cm (negative streamers) [20].

From a macroscopic point of view, the corona discharge on the surface of a conductor can be viewed as an increase of the capacitance of the conductor, while the inductance remains constant due to the low conductivity of the corona region. The capacitance can be estimated from the *q-v* curve, which can be obtained by experimental tests or by models in literature. An example is proposed in Figure 9 of Reference [12].

The lower line of Figure 9 of [12] represents the *q-v* curve measured during the increase of the voltage (i.e., when *dv*/*dt* > 0), while the upper line represents the *q-v* curve measured during the decrease of the applied voltage. In this second phase, the slope, that is, the capacitance, is constant and equal to the geometrical capacitance of the line because it is well-known that corona discharge occurs only when the applied voltage derivative is positive. The lower line presents an increase of the slope with applied voltages higher than 130 kV, which denotes the occurrence of the corona discharge

### **3. Implementation of Corona Discharge**

This section shows the implementation of the corona effect in the procedure proposed in Reference [5] following the one in Reference [12].

The main idea of this approach is to consider a dynamic capacitance in the Agrawal model. As in Reference [12], the dynamic capacitance has been described through the following equation, which aims at reproducing the experimental *q-v* characteristics.

$$\mathbb{C}\_{\text{dyn}}(\mathbf{x},t) = \begin{cases} \mathbb{C}\_{0} & \text{for } u(\mathbf{x},t) < u\_{th}(\mathbf{x},t) \\ \mathbb{C}\_{0} \frac{(k\_{1} + k\_{2}(u(\mathbf{x},t) - u\_{th}(\mathbf{x},t)))}{u\_{th}(\mathbf{x},t)} & \text{for } u(\mathbf{x},t) \ge u\_{th}(\mathbf{x},t) \text{and } \frac{du(\mathbf{x},t)}{dt} > 0, \end{cases} \tag{4}$$

where *k*1 > 1 is related to the sudden change of the capacitance when the voltage exceeds the corona threshold *uth* and *k*2 > 0 is related to the gradual increase of the capacitance when the voltage is rising above the threshold. *C*0 is the geometrical capacitance of the line and *<sup>u</sup>*(*<sup>x</sup>*, *t*) is the voltage on a generic

conductor at time instant *t* and at a distance *x* from the beginning of the line. The voltage threshold *uth* is related to *C*0, to the conductor radius *r* and to the critical electric field in (1), according to

$$
\mu\_{th} = \frac{2\pi\varepsilon\_0 r}{\mathcal{C}\_0} E\_c.\tag{5}
$$

Here, as usual, *ε*0 is the electric permittivity in vacuum.

#### **4. Corona Effect Influence on the Lightning Performance**

This section shows the enhancement of the number of dangerous lightning events due to the corona effect.

Let us consider a single-phase overhead distribution line whose details are available in Table 1. In order to avoid reflections, the line extremities are matched. Moreover, let us suppose a ground conductivity of 1 mS/m and a ground permittivity of 10.


**Table 1.** Line details.

According to Reference [6] and supposing a lightning channel height of 8 km and a propagation velocity along the lightning channel of *c*0/2 (being *c*0 the light speed in vacuum ), the lightning performance of the overhead distribution line is computed as follows (for further details check Figure 1):


$$F = 200 \frac{m}{n\_{tot}} GFD y\_{max} \tag{6}$$

where *GFD* is the ground flash density expressed as number of flashes per square kilometer per year ad *ymax* is the maximum value of the y-coordinate where the events are extracted. According to Reference [1], *ymax* is a function of the CFO and it is computed through the extended Rusck's formula [25], choosing as lightning current the maximum value (*Imax*) obtainable from the probabilistic density function.

$$y\_{\max} = \frac{38.8 I\_{\max} \left( h + \frac{0.15}{\sqrt{\sigma\_x^\*}} \right)}{\text{CFO}},\tag{7}$$

where *<sup>σ</sup>g* is the ground conductivity.

For what concerns the corona effect, in the following we assume *m* = 0.8, *k*1 = 1.2 and *k*2 = 4.8 [12]. Consequently, *Ec* = 35.49 kV/cm and *uth* = 150 kV.

**Figure 1.** Flowchart of the lightning performance computation.

Figure 2 shows the number of dangerous events per 100 km of line per year with the corona effect (solid black line) and without (red dashed line). As can be easily seen, when we deal with overhead distribution lines characterizred by very low CFO (≤100 kV), the corona discharge does not increase the number of flashovers. On the other side, when the CFO is higher, the influence of corona effect is meaningful. This result is in agreemen<sup>t</sup> with the conclusions of References [12,17], which have highlighted an increase of the lightning-induced voltages due to the corona effect. In particular, the enhancement can be caused by the decrease of the wave propagation velocity due to the increase of the line capacitance.

In order to quantify the importance of the corona effect, again in Figure 2 the lightning performance without introducing corona effect and considering a perfect conducting ground (cyano solid line) has been proposed. The percentage enhancement of the number of flashovers due to corona

is comparable with the one caused by considering a lossy ground (Table 2) especially when the CFO is high—when the CFO overcomes 250 kV the dominant effect is ascribed to the corona discharge.

**Figure 2.** Number of dangerous events on an overhead distribution line. Comparison between perfect ground, lossy ground and lossy ground with corona discharge.

**Table 2.** Percentage enhancement—the second column describes how the lossy ground enhances the number of flashovers with respect to the PEC ground, while the third column describes how the corona effect enhances the number of flashovers with respect to the lossy ground case.

