## 1 Introduction  
Renewable energy, such as photovoltaics and wind power, has been widely utilized in power systems to tackle environmental crises and to sustain economic development during the past few decades [1, 2]. Iceland and Norway obtain essentially all of their electricity from renewable sources, and other nations and regions are evolving towards 100% renewables [3]. According to the 2018 BP energy outlook, sustained support for renewables would lead to strong growth in the share of renewables, reaching 25% of total energy consumption by 2040 [4]. Therefore, conventional power systems are evolving towards high penetration of renewable generation. However, the characteristics of renewable generation (RG) are incompatible with traditional power plants. On the one hand, RG (such as solar photovoltaic and wind power) is commonly integrated through voltage-sourced inverters (VSIs), which are almost inertia-less due to their fast switches. On the other, renewable generation profiles, influenced by meteorological disturbances, are stochastic and fluctuant. Correspondingly, RG integration brings great challenges, including frequency stability and reliability. The frequency stability of the power system is highly associated with the inertia property, which is a key indicator for evaluating system strength. The inertia property could slow down the rate of change of frequency (ROCOF) after a frequency event [5]. With increasing penetration of renewable generation and the consequent displacement of traditional synchronous generators (SGs), the overall inertia decreases. The reserve is a crucial issue for sustaining reliability, which ensures the generators' schedules withstand the uncertainty. The system reserves (both upward and downward reserves) could prevent underfrequency load shedding and renewable generation spillage. As renewable generation usually operates at fixed power points for maximum exploitation (maximum power point tracking, MPPT), the system reserves could only be provided by traditional SGs. If the renewable generation reaches a considerable share of the total installed capacity, the traditional plants, which are the only sources of reserves, may not satisfy the reliability criteria [6]. For the Ireland electricity system, the study has shown that more reserves are needed as the penetration level of the wind power increases or the system reliability decreases [7]. Due to direct coupling between mechanical and electric parts in synchronous machines, the inertial response (IR) is naturally provided to suppress ROCOF and mitigate frequency nadir (or peak). Many grid codes have also required inverter-based generation (IBG) to provide IR as an ancillary service [8]. By modifying the supplementary controllers of power electronic interfaces (PEIs) to emulate the external characteristics of traditional SGs, studies have proposed the concept and several approaches for virtual synchronous generator (VSG) algorithms [9; 10; 11; 12; 13]. By sensing circuit parameters (instantaneous voltages and currents) and calculating the instantaneous power, the VSG algorithm enables PEIs to mimic the swing equation, which provides instantaneous inertial response (IIR) [9]. Furthermore, the emulation of the speed governor can also be embedded in supplementary control loops to perform primary frequency response (PFR) [14; 15; 16; 17]. It is noted that, for PEIs under VSG control, energy headroom in renewable units is a prerequisite during frequency response. The energy could come from the kinetic energy in wind turbines or rotors, the electrostatic energy in capacitors, or the stored energy in accessory equipment [1]. However, the above studies are all restrained in the stages of the short-term frequency response [18]. The frequency control of the conventional power systems includes a hierarchical structure, because the system operation requires the ability to respond to change in demand and supply in multiple temporal stages [19].