### History and State of the Art of ESS-Based VSM  
The research of VSM in the early stage focused mainly on the control architecture. Though varying in structure, the basic functions can all be treated as equivalent or similar to droop control, by proper control block transformation [9]. [PERSON] and [PERSON] first proposed the concept of the virtual synchronous machine. Based on the stator emulation equation, the control architecture can be further categorized into two types: voltage to current or current to voltage [10]. Higher-order models of synchronous generators can also be emulated [11]. A VSM model with first-order droop characteristic was adopted in the Europe virtual synchronous (VSYNC) project [12]. As PLL is needed to measure system frequency, this type of VSM is essentially a current source and cannot operate independently. [PERSON] et al. proposed a synchronverter that can emulate both rotor torque and stator voltage characteristics of a synchronous generator with a second-order model [13]. Another VSM topology starts directly from the swing equation [14]. The governor control emulation is presented as a first-order lag element and the stator impedance is also synthesized. Compared with the above VSM concepts, active power is used instead of torque in the swing equation formulation [15]. Grid-forming droop control was proposed as one core control option in microgrid [16]. Assuming inductive grid impedance, frequency, and voltage magnitude can be drooped with output active power and reactive power, respectively. Often a low-pass filter is added after the active power measurement of droop control, making it equivalent to the swing equation but easier to implement than the higher-order-based VSM [17]. Most of the VSM control concepts reviewed above assume an ideal DC bus, which is problematic for practical VSM operation, as most energy resources such as renewable generators adopting maximum point power tracking (MPPT) control, are non-dispatchable and normally do not have sufficient energy reserve for frequency support. By contrast, energy storage system (ESS) with bidirectional power regulation capability is the first choice in VSM realization [10]. Though there are various types of ESS for VSMs, the high cost, limited energy capacity, and response speed of ESS remain as the primary concerns [18]. The amount of energy within ESS, usually measured by state of charge (SoC), must be kept within a reasonable range for secure and sustainable operation. To prevent over-charging/discharging, an SoC feedback control for ESS-based VSM is proposed in [PERSON] et al. 's report [19], but such control cannot ensure constant energy reserve, since it is triggered by SoC-limit violation signals. As reported in [PERSON] et al. 's research [20], the capacity and action time of ESS for a single grid-connected VSM are determined in over-damped, under-damped and critically damped scenarios. However, the theoretical analysis is not applicable if the frequency control on the grid side is considered, for example, in islanded microgrids. An enhanced VSM control for short-term ESS integrated with a type-IV wind turbine generator is proposed in [PERSON] et al's research [21], where proportional-integral (PI) control is added to manage the SoC of ESS. However, the PI control is implemented on the machine-side converter of the wind turbine generator. Therefore, the PI parameters should be adjusted continuously in different rotor speed ranges. Specifically, the PI parameters should be positive or negative based on the signs of slopes of the wind speed and power characteristics. In [PERSON] et al. 's work [22], the constraints of DC voltage deviation, load disturbance, and maximum frequency deviation are considered in inertia synthesis. Nonetheless, the load disturbance is very small (3-5%) to justify the use of a DC capacitor for VSMs. The same authors proposed a VSM control based on hybrid ESS, in which the battery tackles the slow-varying dynamics while the supercapacitor supplies fast-varying components [23].