## 1 Introduction
### Power Reserve Control (PRC)
The inertial-droop approach allows for a very fast inertial control as well as an active power support more extended in time. An extended explanation of this kind of control is given in Section 2, since is the one adopted in the present work. The droop concept for VSWTs is analogous to the synchronous generators one. There are two main types of droop control: fixed or variable droop value. The second is more accurate in responding to frequency deviations, since it depends on wind and power reserve conditions [25] or on the RoCoF [24; 26]. Indeed, it ensures improved system stability and avoids reserve exhaustion. The drawbacks of this kind of control are intrinsic to the physical behavior of the rotor during frequency oscillations. In fact, when the rotor slows down, it has always to respect aero-mechanical limits and avoid stall, keeping a safety margin; consequently, the control is limited, especially for low wind speeds. Furthermore, while the operational life in normal operation has been extensively tested at industrial levels, the operation with inertial control enabled has not, leading to the prescription of a practical utilization of this kind of control in only a few situations by the grid codes [20]. Another issue is related to the fact that, to re-accelerate to optimal speed, the power released to the grid has to be lower than the one coming from the wind. This implies that, for a certain time interval, the power output  
Figure 3: Active power frequency control research lines in wind-based power plants. of the turbine is lowered to recover kinetic energy, delaying frequency recovery and, in some situations, requiring an extra number of reserves to prevent a \"double-dip\" in system frequency, increasing the risk of triggering protective relays at the substation level and causing blackouts [27]. In normal operation, wind turbines follow the MPP to extract the maximum energy available from the wind, achieving the highest aerodynamic efficiency possible. However, if an underfrequency event happens (i.e., the load increases its power), with such an operation there is no margin to respond to the frequency variation. On the other hand, through de-loading, the operation is shifted from the maximum power point to a sub-optimal one, keeping a power margin to face that event [28]. To work in a de-loaded condition, it is sufficient to reduce the coefficient of performance of the turbine (\(C_{p}\)), which is a function of the tip-speed ratio (\(\lambda\)) and of the pitch angle (\(\beta\)). The tip-speed ratio is proportional to the rotor rotational speed (\(\omega_{r}\)). Consequently, it immediately derives the two possibilities to de-load the wind turbine: working in a rotational overspeed condition (as in [29]) or through a pitching control. Usually, rotor overspeeding is applied when the rotor speed is low, while for a rotor speed equal or higher than the rated one the pitch control technique is used [30]. The de-loading of wind turbines has some limitations that historically confined it to very few practical applications. First of all, similarly to what happens in the power curtailment of photovoltaic plants, the de-loading leads to a reduction in the annual capacity factor of the turbine. At the same time, overspeeding can reduce the turbine's life. To determine the power set-point, accurate wind speed measurements are important, otherwise incorrect or rough estimations can highly affect power output and turbine life [20]. Model for a Small Island Power System with Renewable Generators Participating in the Primary Frequency Control  
After an overview of the techniques that allow the considered renewable generators to participate in the inertial and primary frequency control, it is important to go through the model developed in this work. The negative effects on stability due to a massive presence of variable renewable sources are more severe in small and isolated power systems, due to a lack of interconnections with other systems and to their limited inertia.