The effects of friction forces on transient stability

Abstract

The voltage regulators, speed governors, stabilizing control systems, field circuits and damper windings of the synchronous machines determine a dissipation of the energy of a multimachine power system. Such dissipation has a different nature from that caused by the system loads and by the system transfer conductances. In fact it causes a damping of the electromechanical oscillations around the steady-state (while the dissipation due to the system loads and transfer-conductances causes a variation of the pulsation of the oscillations). The problem is: in which way this damping affects stability in the large (that is to say the system behaviour after a large disturbance)? In particular, the introduction of PSS systems, while it improves ‘stability in the small’, could lead to a worsening of ‘stability in the large’. To study this problem we use the direct method of Lyapunov and show the existence of a Lyapunov function that, but for a factor equal to the synchronous speed, represents the conservative part of the system energy. The time derivative of this function is equal to the dissipated power along the trajectories. The stability condition is: the system conservative energy has a minimum at the equilibrium point and, simultaneously, the dissipated power has a maximum. We illustrate a modified equal area criterion from which we deduce that an excessive damping of the steady-state electromechanical oscillations can have a negative effect on transient stability (i.e. stability in the large). The above results are formulated in a general way, that is with regard to a dynamic system with n degrees of freedom, subjected to conservative positional forces and to dissipative forces linearly dependent upon the speed (‘friction forces’). Therefore, they find application in various fields of Physics and Engineering.