## 2 Description of the Developed Methos  
In this section, the method used to find the expression of the maximum sag in a generic span as a function of the angle at the suspension clamp is explained. The taut cable has been modelled as a parabolic function, which is known to be sufficiently accurate for engineering purposes, excluding some special cases such as very long crossings [19]. The general case of a span with length \(l\) characterized by two supports at different levels, in which the height difference denoted as \(h\) (Figure 1), is considered. The angles at the suspension clamps are \(\theta_{1}\) and \(\theta_{2}\), while the maximum sag is \(f_{M}\). Note that in the case of a span with supports at unequal levels, the maximum sag does not coincide with the vertex point of the parabola. The reference frame zero is positioned at the parabola vertex point, and \(x_{1}\) and \(x_{2}\) represent the horizontal coordinates of the supports A and B with respect to the reference system origin. The first step is obtaining the expression of the maximum sag, which is the maximum vertical distance between the generic parabolic function \(y_{P}\) (Equation (1)) and the straight line \(y_{R}\) (in red) connecting points A and B of the parabola (Equation (2)). \[y_{P}=kx^{2}, \tag{1}\]  
\[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]  
where \(k\) is a generic constant to take into account the mechanical parameters of the conductor. Referring to Figure 1, the relationship between \(x_{2}\) and \(x_{1}\) coordinates is given by Equation (3). \[x_{2}=x_{1}+l, \tag{3}\]  
Figure 1: Schematization of the span with unequal level supports with a parabolic function. Angles are positive when describing a counterclockwise rotation. The sag\(f(x)\), as a function of the horizontal position, is then expressed as the difference between \(y_{P}\) and \(y_{R}\) (Equation (4)). \[f(x)=y_{P}-y_{R}=kx^{2}-k(x_{2}+x_{1})x+kx_{2}x_{1}, \tag{4}\]  
The goal is to find the maximum of the sag function \(f(x)\), therefore its derivative with respect to the \(x\) coordinate must be equal to zero. In this way, the coordinate of the maximum sag \(x_{M}\) could be found as indicated in Equation (5). \[x_{M}=\frac{x_{2}+x_{1}}{2}, \tag{5}\]  
As can be appreciated, the maximum sag location is in midspan also in the case of a supports with different heights. By substituting Equation (5) in Equation (4) and making some rearrangements, a generic expression of the maximum sag \(f_{M}\) could be obtained (Equation (6)). \[f_{M}=\frac{kl^{2}}{4}, \tag{6}\]  
At this point, the target is to express \(f_{M}\) as a function of one angle at the suspension clamps \(\theta_{i}\) and of the difference between suspension height \(h\). The tangent of the angle at the span extremity \(\theta_{i}\) is obtained by evaluating the derivative of the parabolic function in correspondence of the \(x_{i\ (i=1,\ 2)}\) coordinate (Equation (7)). \[\tan\theta_{i}=\left.\frac{dy_{P}}{dx}\right|_{x_{i}}, \tag{7}\]  
Using Equation (7), the expressions of \(x_{1}\) and \(x_{2}\) are thus obtained as functions of \(\theta_{1}\) and \(\theta_{2}\) (Equation (8)). \[x_{1}=\frac{\tan\theta_{1}}{2k},\ \ x_{2}=\frac{\tan\theta_{2}}{2k}, \tag{8}\]  
Next, the height difference \(h\) can be computed as the difference between the two parabola points evaluated at the support positions (Equation (9)). \[h=k(x_{2}^{2}-x_{1}^{2}), \tag{9}\]  
By substituting the relationship between the \(x_{2}\) and \(x_{1}\) coordinates (Equation (3)) in Equation (9) and taking advantage of Equation (8), it is possible to express the constant \(k\) as function of the height difference \(h\) and one of the two suspension angles \(\theta_{i}\), obtaining the two expressions in Equation (10). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]0  
In the end, the expression of the maximum sag\(f_{M}\) as function of \(h\) and \(\theta_{1}\) is obtained in Equation (11) by substituting Equation (10) in Equation (6).