## 1 Introduction
### Inertial Response
#### 3.2.2 Measurement and Quantification  
Supposing that a DRES can exhibit inertial response that corresponds to an inertial time constant \(H\), then (7) can be used to evaluate this response, \(p_{H}=p_{m}-p_{e}\) for a given ROCOF, \(df/dt\). Such a response is shown in Figure 8 for a PV system that incorporates in its DC a supercapacitor bank able to provide inertial with \(H\) = 5 s. The physical configuration of the system under study is presented in Figure 3. More specifically, the PV system is directly connected to the upstream network, which is modelled as an ideal voltage source in series with an impedance. Initially, the PV system injects 0.25 pu to the grid, while the network frequency remains constant. Afterwards, to investigate how the PV system reacts on frequency disturbances, the network frequency is modified by varying the frequency of the ideal voltage source. There are three periods with negative, zero and positive ROCOF. Both the positive and the negative ROCOF are equal to 0.02 pu/s (1 Hz/s), as shown in Figure 8.  
Note that a new model for the grid-interfaced converter of the PV system has been adopted for these simulations. It has been developed in frame of the EASY-RES project and is called \"Unified Virtual Synchronous Generator\" (UVSG). Scope of the UVSG model is to incorporate a series of functionalities that will allow the widespread penetration of renewable energy sources in both distribution and transmission systems. Among them, the provision of inertia response is regarded as one of the most important functionalities that improves the frequency stability of the grid. More details regarding this model are presented in [28]. The inertial response derived from the UVSG model and shown in Figure 8 can be confirmed with (7) since, \(p_{H}=2H\frac{df}{dt}=2\times 5\times 0.02=0.2\) pu, resulting in a step increase of the electrical power from 0.2 pu to 0.4 pu. In such cases with high ROCOF, thereby high \(p_{H}\), the measurement and quantification of the inertial response can be evaluated from the sudden increase (or decrease) of the electrical power of the DRES for as long as the ROCOF lasts. However, during most of the time, the frequency deviates with very small ROCOF as shown by the frequency measurement for a time interval of 10 s with 20 ms resolution in Figure 9. The frequency measurement was made on 23 August 2018 by the Greek TSO on a 150 kV bus near Athens, Greece, via a PMU. Figure 8: Simulation of inertial response of a PV system with H = 5 s, under ROCOF = 0.02 pu (1 Hz/s). In such cases, the electrical power of a DRES will exhibit small variations too, making the measurement of inertial response questionable. Figure 9 shows the electrical power, during the night, of a PVPP with 20 kVA nominal inverter power, which is controlled to provide inertial response with H = 9.5 s. It can be seen that the electrical power, which in this case is the result of the inertial response, varies as little as 100 W or 0.5% of the nominal power. Such small quantities may fall within the measurement accuracy even of smart meters. On the other hand, if we assume that the frequency pattern in Figure 9 is repeated for a whole year, the inertial response would sum up to 200 kWh which is something tangible and cannot be neglected. Based on the aforementioned obstacles, we suggest that the quantification of inertial response by a DRES is based on calculations as follows: it is evident from (7) that the inertial response depends only on \(H\) and ROCOF. The value of \(H\) should be validated in a certified lab by subjecting the DRES converter to tests similar to the one shown in Figure 8. The ROCOF can be evaluated in discrete time steps by measuring the frequency and time in a window of 100 ms as suggested in the recent IEEE 1547-2018 standard [10]. Thus, the inertial contribution \(E_{i}\) expressed in terms of energy over a period \(T\) (an hour, day, etc.) can be calculated by  
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