This paper considers the task of attitude control of an EDTS moving in the near-Earth space in the position where the tether is tensioned along the local vertical. Analytical studies and computer modeling indicated that the Lorentz moment acting on the EDTS due to charged collectors on the tether ends significantly increases the stability margin for the vertical position of the tether. Besides, the paper demonstrated the possibility to generate control moment of the dissipative character by actively controlling the charge value according to the conditions determined by the current tether orientation. If the tether parameters are chosen according to the conditions taking into account the effect of disturbing moments, the asymptotic stability of the tether orientation along the local vertical is achieved without having to switch off the current running in the tether. The proposed device and the control method thereof can be used for stabilizing the EDTS in the near-Earth space for enhancing its operating efficiency in the process of space debris collection.