The goal of this virtual lab is to show the importance of a stabilizing control action in power networks.

In order to work properly, power networks require that the frequency does not deviate too much from a nominal value, i.e. the standard 50Hz. The so called "frequency deviation" \(\Delta f\), which is the difference between the actual frequency and the nominal frequency, has to be then always very close 0, otherwise a failure of the system will occur, which in turn implies a blackout for the end users.

Systems that are not properly controlled are not able to drive the frequency deviation towards 0 and are therefore prone to failures. Engineers have studied during the year many possible solutions to reduce the frequency deviation in power systems during the years. One approach is to model the system with ordinary differential equations that can show how the system evolves. Using tools from control methods, one can determine whether the system is going to be:

- unstable, which means that the state trajectories will diverge and the system will not converge to its desired setpoint;
- simply stable, which implies that the state trajectories will stay in a neighborhood of the desidred setpoint;
- asymptotically stable, which, as the name says, means that the system will asymptotically converge to the desired setpoint.

More stability definitions exist, but the explanation is beyond the scope of this virtual lab.

It is possible to synthethize a controller that stabilizes an unstable system, or that makes faster the convergence to the desired setpoint of a slow system. The approach that we present here is called state-feedback control and it is based on using the available information on the system to compute a control input, which will be proportional to the measured state. We do not want to focus on the equations here as the overall goal of this virtual lab is to show how important a good controller is for power networks.

The first slider can be used to set the value to the initial disturbance on the system and has to be used before pressing the "play" button. The Button has the function of enabling or disabling the controller. Modify the initial condition from 0 to another value and simulate the system with the controller enabled and observe the behavior. The several colored plots show the different frequency deviations in the four areas of the model. Can you note that the value of the frequency deviations keep growing and growing? This happens because the system is unstable and no stabilizing controller acts on it, therefore the system diverges. A small perturbation, or disturbance, is enough to destabilize the system. Indeed, if we start the simulation again keeping the initial disturbance equal to 0, the system will not diverge, since the initial conditions keep the frequency deviation constant at an equilibrium point. This equilibrium, however, is unstable, as a small perturbation will destabilize the system. The concept is better shown with the picture below: point A is an equilibrium point, but it is unstable, because a small disturbance on the ball will make it roll, while point B is a stable equilibrium point, because the ball will go back to the same even if perturbations are present.

Now, restart the simulation and this time enable the controller (once again modify the initial condition). Observe how the control action can lead all the deviations towards zero, i.e. towards stability, in an asymptotic way.

Lastly, restart the simulation once again, and enable the controller. When the simulation is still running, destabilize the system by disabling the controller. Can you observe how the frequency deviation starts growing in module again? This is due to the fact that there is no controller now that can compesate the instability of the system. Without stopping the simulation, enable again the controller, and see how the system is stabilized again. Repeat the operation several times to observe how the controller is always able to stabilize again the system.

Control of a multi-area network

Tomas Pippia