Open PowerWorld Simulator case Problem 12_3. This case models the system from Example 12.1 except with the rate feedback gain constant, \(\mathrm{K}_{\mathrm{f}}\), has been set to zero and the simulation end time was increased to 30 seconds. Without rate feedback the system voltage response will become unstable if the amplifier gain, \(\mathrm{K}_{\mathrm{a}}\), becomes too large. In the simulation this instability will be indicated by undamped oscillations in the terminal voltage (because of the limits on \(V_{r}\) the response does not grow to infinity but rather bounces between the limits). Using transient stability simulations, iteratively determine the approximate value of \(\mathrm{K}_{\mathrm{a}}\) at which the system becomes unstable. The value of \(\mathrm{K}_{\mathrm{a}}\) can be on the Generator Information Dialog, Stability, Exciters page.

Example 12.1

Assume the two-axis generator is augmented with an IEEE Type 1 exciter with \(\mathrm{T}_{\mathrm{r}}=0, \mathrm{~K}_{\mathrm{a}}=100, \mathrm{~T}_{\mathrm{a}}=0.05, \mathrm{~V}_{\mathrm{r} \text { max }}=5\), \(\mathrm{V}_{r \text { min }}=-5, \mathrm{~K}_{\mathrm{e}}=1, \mathrm{~T}_{\mathrm{e}}=0.26, \mathrm{~K}_{\mathrm{a}}=0.01\) and \(\mathrm{T}_{\mathrm{f}}=1.0\).

(a) Determine the initial values of \(\mathrm{V}_{r}, \mathrm{~V}_{f}\), and \(\mathrm{V}_{\text {ref }}\).

(b) Using the fault sequence from Example 11.10, determine the bus 4 terminal voltage after 1 second and then after 5 seconds.