###### Abstract  
Renewable energy resources (RES), e.g., solar photovoltaic (PV), wind energy, and electric vehicles (EVs) are extensively integrated into modern power electronics-based power systems due to their sustainable and environmentally friendly features. This has gradually reshaped the frequency regulation structure of power systems. Specifically, the duty of frequency regulation, previously supported by synchronous generators, will be taken over by grid-tied power converters (such as EV chargers), which can be controlled as virtual synchronous machines (VSMs) to mimic the inertial response of synchronous generators. However, being different from synchronous generators, VSMs enable fast frequency regulation. As a consequence, it is identified in this paper that the elimination of frequency oscillations is possible. Therefore, instead of simply increasing inertia for oscillations suppression, the desirable inertia coefficient can be different for various cases. In order to illustrate this point, relationships between several important performance indices, such as the rate-of-change-of-frequency, frequency nadir, overshoot, settling time, and inertia coefficient, are analyzed in detail. Furthermore, a straightforward method for designing the optimal inertia coefficient for VSMs is put forward. Finally, experiment results obtained from a 1-kW three-phase VSM prototype are demonstrated to validate the effectiveness of the theoretical analysis and proposed design method. Electric vehicle (EV); frequency regulation; renewable energy resources (RES); power system inertia; virtual synchronous machine (VSM) +
Footnote †: journal: Applied Sciences  
frequency will vary until the system reaches another steady state of power balance. For traditional power systems, synchronous generators absorb or release their rotational kinetic energy during frequency dynamics to help slow down the frequency change. In order to quantify this effect, the concept of inertia is proposed, and it is defined as the ratio of kinetic energy to the rated power of synchronous generators [5]. In modern power electronics-based power systems, synchronous generators are gradually replaced by grid-tied dc-ac power converters (inverters). However, since these converters normally operate to extract the maximum power from RES without absorbing/providing any kinetic energy, the system inertia contributed by synchronous generators continues to reduce [6]. This leads to larger frequency deviations and changing rates during frequency events, which may further cause undesirable loading, shedding, or even blackouts [7]. One effective approach to increase the system inertia during frequency events is the adoption of energy storage systems for inertia emulation [8]. In [7], an additional energy storage system consisting of supercapacitors was employed to provide frequency support in electrical islands with large shares of RES. After a generation outage, i.e., a frequency event, the energy storage system promptly injected power into the grid, leading to reduced system frequency deviations and changing rates. Similarly, a supercapacitor-based energy storage system was incorporated in a wind generation system to produce virtual inertia for compensating frequency disturbances as well as intermittency caused by wind fluctuations in [9]. Despite those additional energy storage systems allowing a fast reaction to power imbalance, this approach brings other concerns, e.g., increased system complexity and maintenance costs. Another interesting attempt is to utilize existing energy storage units in EVs for frequency regulation. With a desirable controller, grid-tied converters in EVs can contribute grid services, known as vehicle-to-grid (V2G) services. When operating as synchronous generators, grid-tied converters are known as virtual synchronous generators (VSG) [10], virtual synchronous machines (VSM) [11], or synchronverters [12]. Notably, when imitating synchronous generators, grid-tied converters can be controlled to achieve different levels or extents of emulation.