## 4 Simulation Results
### Load Power Step Up/Down  
The first and simpler test to analyze the frequency response of a dynamic system is obtained by observing its behavior after a step perturbation. In this case, the step is an instantaneous change of the load power (\(P_{load}\)). This situation is extremely simplified with respect to what happens during the normal operation of a power system, but nevertheless this test is really important to take a first look at the effectiveness of the ability of the primary frequency control in stabilizing the frequency. In the small island system that is described here through an LFC approach, a sudden step change of the load power is more likely to happen as a reduction. In other words, it is more common that the load decreases suddenly (for example due to a disconnection of an industrial user) than the vice versa. This is why the maximum amplitude tested for the steps in the reduction is higher than the one in the step up of the load power. The first plot reported regards Simulation 1, in which the load suddenly increases its power by 1%. Figure 10 shows the frequency oscillation of the system due to the consequent imbalance between generation power and load power. First of all, it can be observed that an increase in the load power causes a decrease in the frequency. In Figure 10, the frequency nadir can be easily individuated as the minimum of the frequency in the time plot. Observing it in the four scenarios: as expected, in the absence of frequency response capabilities from the renewable sources (w.o.FC), it is lower than in the other three scenarios. Indeed, it is equal to 0.9983 pu (49.915 Hz) against the 0.9990 pu (49.95 Hz) in case all the generators participate in the primary control (cFC). It is evident how the intervention of wind and photovoltaic generators in stabilizing the frequency is helping the system, since they reduce the magnitude of the nadir and, after a few instants, they also reduce the oscillations of the system's frequency and the stabilization time. The superposition of their effects lead to the behavior in purple, which is one of the scenarios in _complete FC_, i.e., all the generators participate in the primary frequency control. Figure 10: Simulation 1 frequency trend. Of course, since this work focuses just on the primary (and inertial) frequency control, the final value of the frequency is not the nominal one, that requires also the actions of the secondary and tertiary control to be restored. Looking at the RoCoF plot in Figure 11, it can be seen that the PV system produces oscillations that increase the RoCoF in proximity of the step. This is a negative fact, which would require an additional PRRC to be solved, that reflects also on the _complete FC_ scenario. After the peak of the RoCoF, anyway, the scenario for which it is higher is the one without frequency control. This allows for the understanding that the PRC for the PV system is capable of containing the frequency and the RoCoF in the instants after a sudden variation in the load power. On the other hand, the wind virtual inertia operates very well both during and after the load power step. Remarkably similar results are obtained from Simulation 2 as shown in Figure 12: a negative step of load power of the same amplitude produces the same behavior of the frequency but mirrored to an increase in frequency. The nadir is also simply mirrored upwards with respect to Simulation 1: in the scenario with the renewables with disabled frequency control, it is equal to 1.0017 pu, while for the _complete FC_ it is equal to 1.0010 pu. Figure 11: Simulation 1 RoCoF. Figure 12: Simulation 2 frequency trend. In the next simulations, larger steps are introduced. Due to the similarity already discussed between the first two, they are expected to produce a frequency waveform with the same shape but a different amplitude (dilated by a factor of 2 and 10, respectively). Looking at the results for Simulation 3 shown in Figure 13, this expectation is confirmed.