2. Transmission line equation of motion (EOM)

The transverse displacement of high voltage transmission line conductor is generally caused by wind loading. This form of vibration with small displacement is known as aeolian vibration and it is a source of concern to the power lines reliability. One the vulnerabilities is that it can cause fatigue failure of the transmission lines. Conductors are example of continuous or distributed systems and modeling its mechanical vibration can either be as a beam or taut string. In [18, 19], it was ascertained that modeling a conductor as a beam is more accurate than modeling it as a taut string due to the effect of the bending stiffness. Hence, in line with the above, the conductor transverse vibration was modeled as a beam, simply supported or pinned at both ends. The distributed loading on the conductor is replaced by effective point load that can effectively have the same resultant effect as that of the actual distributed load.

The high voltage transmission line equation of motion was formulated by assuming that power conductors can modeled as beams with fixed ends. The following assumptions were considered [1]:


The assumptions were based on beam theory. In considering the power conductor as a beam, sagged by a tensile force S, being acted upon by a concentrated wind load f xð ; tÞ, with crosssectional area A, density r, flexural rigidity EI, displaced at a distance of x after time. In Eq. (1), the high voltage transmission line equation of motion is expressed as:

$$f(\mathbf{x},t) = EI \frac{\partial^4 y(\mathbf{x},t)}{\partial \mathbf{x}^4} - S \frac{\partial^2 y(\mathbf{x},t)}{\partial \mathbf{x}^2} + \rho A \frac{\partial^2 y(\mathbf{x},t)}{\partial t^2} \tag{1}$$

For x∈ ð0; 1Þ, t > 0. The boundary conditions are expressed and indicated in Eqs. (2) and (3):

$$y(0,t) = \frac{\partial^2(0,t)}{\partial x^2} = 0\tag{2}$$

$$y(l,t) = \frac{\partial^2(l,t)}{\partial x^2} = 0 \tag{3}$$

The initial conditions at t ¼ 0 are indicated in Eq. (4), Eq. (5) and expressed as:

$$y(\mathbf{x},0) = y\_o(\mathbf{x})\tag{4}$$

$$
\dot{y}(\mathbf{x},0) = \dot{y}(\mathbf{x})\tag{5}
$$

Introducing the mass per unit length of the power conductor, the new equation of motion indicated in Eq. (6) is expressed as:

$$f(\mathbf{x},t) = EI \frac{\partial^4 y(\mathbf{x},t)}{\partial \mathbf{x}^4} - S \frac{\partial^2 y(\mathbf{x},t)}{\partial \mathbf{x}^2} + m \frac{\partial^2 y(\mathbf{x},t)}{\partial t^2} \tag{6}$$

In order to derive a possible solution, the model was simplified using dimensionless functions and Dirac delta functions. In Eqs. (7)–(12), the variables are expressed in dimensionless form and expressed as:

$$Y = \frac{y(\mathbf{x}, t)}{D} \tag{7}$$

$$X = \frac{x}{L} \tag{8}$$

$$
\pi = \frac{t}{f} \tag{9}
$$

$$I\_p = \frac{Df^2}{\mathcal{g}}\tag{10}$$

$$S\_p = \frac{SD}{\chi L^2} \tag{11}$$

$$M\_p \frac{EI}{\gamma L^4} \tag{12}$$

Eq. (13) indicates the revised equation of motion and it is expressed as:

$$M\_p \cdot \frac{\partial^4 Y}{\partial X^4} - S\_p \frac{\partial^2 Y}{\partial X^2} + I\_p \frac{\partial^2 Y}{\partial \tau^2} = \frac{1}{\mathcal{V}} \left[ F(X, \tau) + \sum\_n \delta(X - X\_n) F\_n(\tau) \right] \tag{13}$$

Where γ represents the power conductor weight per unit length and g represents gravitational constant. XnδðX � XnÞ represents the Dirac delta function, F Xð ; τÞ denotes the net transverse force per unit length acting on the conductor and Fnð Þτ denotes the nth concentrated force acting transversely on the conductor.
