Linearize the swing equations around an equilibrium point.
Compute the transfer function from the input U to the output !.
For which equilibria is the linearizes system stable?
Using the equilibrium point ((=6; !s); 1), simulate the linearized system and the non- linear swing equations for di erent initial conditions. Comment on what you observe. Illustrate your observations with 2 plots of your choice.
When a fault occurs in a transmission line the generator will either accelerate or decelerate. We will simulate this by using the initial value for ! to be 1% higher than !s. How long do you have to wait until the system returns to equilibrium? Illustrate your answer with a plot.
Design a controller to reduce by 50% the time to reach the equilibrium (under the initial conditions of the previous question). Show a plot of the evolution of the system when using the designed controller.
Keep increasing the value of ! until the system (swing equation+controller) becomes unstable. Compare with what happens when you do not use a controller.
Redesign your controller so that it stabilizes the angular velocity under faults that change the initial value of ! no more than 10% of its equilibrium value. Illustrate the operation of your controller with the relevant plots.
In addition to maintaining the angular velocity at !s we are also interested in controlling since the value of determines the current that ows through the transmission line to which the generator is connected to. Compute the transfer function for the linearization around ((=2; !s); 1=2). Is the linearized system stable?
Design a controller so that when starting from ! = !s and from a value of that is at most 10% away from =2, reaches =2 in less than 2 seconds with less than 1% of overshoot. Illustrate the operation of your controller with the relevant plots.