Improving Damping to Power Oscillations

The swing equation of a synchronous machine is given by (9.14). For any variation in the electrical quantities, the mechanical power input remains constant. Assuming that the magnitude of the midpoint voltage of the system is controllable by the shunt compensating device, the accelerating power in (9.14) becomes a function of two independent variables, | V_M | and δ . Again since the mechanical power is constant, its perturbation with the independent variables is zero. We then get the following small perturbation expression of the swing equation

(10.12)

where Δ indicates a perturbation around the nominal values.

If the mid point voltage is regulated at a constant magnitude, Δ| V_M | will be equal to zero. Hence the above equation will reduce to

(10.13)

The 2^nd order differential equation given in (10.13) can be written in the Laplace domain by neglecting the initial conditions as

(10.14

The roots of the above equation are located on the imaginary axis of the s-plane at locations ± j ω^_m where

This implies that the load angle will oscillate with a constant frequency of ω_m . Obviously, this solution is not acceptable. Thus in order to provide damping, the mid point voltage must be varied according to in sympathy with the rate of change in Δδ . We can then write

(10.15)

where K_M is a proportional gain. Substituting (10.15) in (10.12) we get

(10.16)

Provided that K_M is positive definite, the introduction of the control action (10.15) ensures that the roots of the second order equation will have negative real parts. Therefore through the feedback, damping to power swings can be provided by placing the poles of the above equation to provide the necessary damping ratio and undamped natural frequency of oscillations.

Example 10.2