## 1 Introduction  
Renewable generation (RG), such as the solar PV (photovoltaic) and wind power, has been widely utilized in the last decades due to the depleted fossil fuels and the environment concerns [1; 2; 3]. Compared to traditional power plants, renewable generation units are more scalable to be either equipped close to the demand or concentrated into renewable plants. Voltage-sourced converters (VSCs) are commonly equipped for the integration of renewable generation, which provides fast and accurate adjustments on generation profiles [4; 5]. The solar PV and wind power can be categorized as inverter-based generation (IBG) because inverters are required to regulate their outputs [6; 7; 8]. According to the BP energy outlook 2018, by 2040, there would be strong growth in the share of renewables, reaching 25% of total energy consumption in an evolving transition scenario [9; 10]. Therefore, with the increasing penetration of renewable generation and the consequent displacement of traditional synchronous generators (SGs), the current power system has been undergoing an intrinsic transition toward converter-dominated grids. However, the fast power electronic interfaces (PEIs) (which are tens or hundreds of times faster than system frequency) also incur problems as they have almost no inertia. The overall inertia is an important indicator to evaluate system strength. On the one hand, when frequency events happen, inertia response (IR) could slow down the change of the range of frequency (ROCOF). On the other hand, it buys time for the activation of the primary frequency response (PFR, which kicks in when the frequency deviation crosses a preset deadband) and mitigates the nadir [11]. In the meantime, the parallel operation involving SGs and the PEI-based RG could cause unstable operation because their responses activate at very different places. Moreover, for the paralleled RG units, autonomous operation and proportional load share are also hard to achieve [12; 13; 14; 15; 16; 17]. The critical problem for high-penetration IBG integration is caused by its incompatible characteristics with the synchronous machines. In other words, the IBG units could not provide IR and PFR jointly and collaboratively with the SGs. As the current power system is still dominated by the synchronous machines, it is desired to make the IBG compatible in current power systems. Under this background, the concept of the virtual synchronous generator (VSG) is proposed [18; 19; 20; 21; 22; 23; 24]. The VSG concept is to modify the highly programmable supplementary controllers of the VSC interfaces and mimic the external characteristics of the traditional SGs [18]. By the VSG control, the emulation of IR and PFR can be implemented [18; 19; 20; 21; 22; 23; 24]. However, as one of the major advantages for power converters is its ability of fast response, the fixed inertia property may limit the response time and flexibility. So, there is a tradeoff between fast response and frequency stability [22; 25; 26]. As the parameters of PEIs are tunable rather than predominated by physical equipment, VSG-IBG units could also alter their external properties for advanced performances. Continuous changes and the step change of virtual inertia parameters are designed to minimize the frequency deviation and energy support [27; 28; 29]. However, the parameter settings are straightforward but arbitrary. In [30], quantitative performance analysis is presented for the inertia response provided by the capacitors in the DC links. From its results, the designs for capacitance and DC voltage are completed. However, the capacitors can only provide short-term inertia response. As the frequency response involves two stages, including the inertial response and the primary frequency response, then the virtual inertia constant, virtual speed droop, and the load disturbances should be considered.