## 2 Description of the Developed Methos
\[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]1  
Analogously, Equation (12) expresses the maximum sag\(f_{M}\) as function of \(h\) and \(\theta_{2}\). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]2  
It is also possible to find the expression of \(f_{M}\) as a function only of the two angles at the span extremities \(\theta_{1}\) and \(\theta_{2}\) by substituting the two expressions of Equation (8) into Equation (3) and rearranging the terms (Equation (13)). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]3Again, substituting the expression of \(k\) in Equation (6), an alternative formulation of the maximum \(\mathrm{sag}f_{M}\) is obtained (Equation (14)). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]4  
In conclusion, in the case of a span with supports at different levels, the maximum sag is function of one angle at the suspension clamp (e.g., \(\theta_{2}\) in the case of Equation (12)) and of the height difference \(h\) between the two supports, or it is a function of both angles at the suspension clamp. Therefore, the maximum \(\mathrm{sag}f_{M}\) can be computed by measuring both angles at the two extremities (Equation (14)) or by knowing the height difference \(h\) between the supports and measuring the angle only at one extremity (Equations (11) and (12)). Considering the case of a span with supports at the same level (\(h=0\)), Equation (12) simplifies in Equation (15). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]5  
Since angles at the suspension clamps can be considered sufficiently small, the tangent of the angle can be approximated to the angle itself and the final expression of \(f_{M}\) is obtained (Equation (16)). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]6  
Therefore, it was demonstrated that the maximum sag in a span with supports at the same height could be computed by measuring one of the angles at the span extremities. The same expression can be obtained more quickly through an engineering approach, explained as follows, always referring to the case of equal support levels. The expression of the maximum sag according to the well-known \"flat parabola method\", which can be found in the literature [7], is given by Equation (17). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]7  
where the cable weight per unit length \(w\) is modelled as a uniform distributed load along the span \(l\) and \(T_{H}\) represents the horizontal component of the tensile load T (Figure 2). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]8 Schematic representation of a conductor span of length \(l\) modelled according to the “flat parabola method”. The vertical component of the tensile load for equilibrium on the span is given by Equation (18). \[y_{R}=k(x_{2}+x_{1})x-kx_{2}x_{1}, \tag{2}\]9  
If the vertical component of the tensile load is assumed to be kept constant along the span, the tangent of the angle at the suspension clamp \(\theta\) can be expressed by the ratio between the vertical and horizontal components of the tensile load \(T_{V}\) and \(T_{H}\) (Equation (19)). \[x_{2}=x_{1}+l, \tag{3}\]0  
By substituting Equation (18) in Equation (19) the expression of \(T_{H}\) can be computed (Equation (20)). \[x_{2}=x_{1}+l, \tag{3}\]1  
Substituting Equation (20) into Equation (17), Equation (16) is again obtained. In conclusion, it was shown that, by means of both methods, a simple and straightforward expression for the sag estimation based on the measurement of one angle at the suspension clamp is obtained for the case of a span with levelled supports. This way, the \(k\) parameter and, consequently, the sag\(f_{M}\), are calculated through geometrical quantities that are easily measurable (angle at the suspension clamp), without the need of the cable self-weight \(w\) and the horizontal component of the tensile load \(T_{H}\), which are parameters difficult to obtain. An example of angular measurements necessary for the estimation on the maximum sag is proposed in Section 3.