## 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 \(E_{c}\).  
\[E_{c}=m\times 2.594\times 10^{6}\left(1+\frac{0.1269}{r^{0.4346}}\right), \tag{1}\]  
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 \({}^{\circ}\)C, a pressure of 760 mmHg and a humidity of the air equal to 11 g/m\({}^{3}\). 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 \(E_{c}\). With respect to (1) an increase of the air humidity to 18 g/m\({}^{3}\) 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.  
\[E_{c}=31.53\left(1+\frac{A}{K^{a}n_{s}^{b}r^{c}}\right), \tag{2}\]  
where  
\[K=\delta^{1.01}\left(1+0.08\left(\frac{H}{11}^{0.72}-1\right)\right), \tag{3}\]  
being \(\delta\) the relative air density, \(H\) the air humidity, \(n_{s}\) 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