In the electric grids of the Russian Federation, the reliable operation of distribution grids with a voltage of 6–35 kV is attracting more and more attention. Their operation is under significant influence of lightning overvoltages associated with a direct lightning strike into the line, and induced overvoltages when lightning strikes the ground near the overhead line. The lightning overvoltages affecting the 6–35 kV overhead lines lead to an increase in their accident rate during operation due to the relatively low impulse electric strength of the linear insulation of the lines. Cable protection against lightning overvoltages is not used for the 6–35 kV overhead lines in the Russian Federation. As calculations show, the installation of a grounded cable on overhead lines due to the coefficients of the connection between the cable and the wire reduces the magnitude of the induced overvoltages. The paper presents a mathematical model for calculating the electromagnetic component of an induced overvoltage based on an approach that uses the wave equation of a line with a voltage source distributed along its length, created by an EMF induced by lightning current in the presence of a grounded wire on 35 kV power lines. The issues of inductive overvoltage arising during repeated lightning discharges, which can lead to large amplitudes of the magnetic inductive component of overvoltages in comparison with the main discharge of the lightning current, are investigated. The electrostatic component of induced overvoltages has lower values during repeated lightning discharges, since the discharge occurs through a partially ionized channel, in which the leader stage of the discharge is characterized by lower electric field strengths. It is shown that the use of a grounded wire with distributed conductivity on a 35 kV overhead line causes a potential on the wire, which at steep fronts of the lightning current is close to 50% of the value of the magnetic component of the induced voltage on the phase wire.