The Mathematical Modeling of a Lightning Strike in an HVAC Line Considering the Modified Hamilton–Ostrogradsky Principle

Based on the modified Hamilton–Ostrogradsky principle, a mathematical model of a distributed-parameter high-voltage HVAC line that includes lightning shield wires is proposed. A partial differential equation of a five-wire power line is produced as a result. Therefore, a methodology for looking for boundary conditions of a long line equation in the five-wire version is proposed here. A mathematical model is introduced as an example of a section of a power line that consists of a high-voltage long line that includes shield wires operating in an equivalent concentrated-parameter power system presented in its circuit version. The system is described with both partial and ordinary derivative differential equations. Poincaré boundary conditions of the third type are applied to solve the state equations of the object discussed. A discrete line model is thus presented, described with ordinary differential equations based on the well-known straight-line method. Transient processes across the system are analysed exactly at the moment of a lightning strike against a shield wire in the middle section of the line. To this end, a mathematical lightning strike model is developed by means of cubic spline interpolation. The original system of differential equations is integrated into the implicit Euler method, considering the Seidel method. The end results of the computer simulation are presented graphically and analysed. The results show the effectiveness of the proposed method of analysing transients across ultra-high-voltage lines that include lightning protection wires and can serve as accurate calculations of power supply lightning protection at the stages of design and production.