## 2 Implementation of VI-Based Inverters
### Vsm  
Both VSM and VISMA are abbreviations for the virtual synchronous machine. However, VSM is more commonly used. Both are based on the emulation of SM but with some slightly different implementation topology. Various researchers refer to different abbreviations, but essentially VSM and VISMA are interchangeable terms. Although the terms are different, all of them shares similar VI emulation techniques. The objective of the VSM is to provide grid forming employing frequency droop and VI. A phase lock loop (PLL) is typically used for frequency estimation to calculate the RoCoF in driving the VI loop [29]. The use of PLL is eliminated during regular operation. It is because VSM can synchronize with the grid based on the power balance after initialization by PLL [30]. VSM uses synchronous reference frame or d-q based mathematical model of an SG. The Park transformation of abc to direct-quadrature-zero (_dq0_) converts three-phase (_abc_) signal to _dq0_ rotating reference frame. The inverse Park transformation (_dq0_ to _abc_) turns _dq0_ rotating reference frame back to three-phase (abc) signal. Generally, the Park transformation increases the complexity of the system and total execution time (TET). The measurements of the grid voltage are used to calculate the stator currents. However, VISMA model exhibits numerical instability. The first VSM concept was proposed  
Figure 6: The VI-based inverter enables the imitation of conventional SG-based power system. Figure 7: The typical connection of a generic VI-based inverter. by [PERSON] and [PERSON] in 2007 and labeled as VISMA [31]. Virtual synchronous machine zero inertia (VSM0H) is a variant of VSM, which is identical to VSM, but with zero inertia constant (H) as the name suggests. Unlike VSM, it operates without the requirement of PLL. The phase angle of the voltage is derived from an integrator which advances at a rate determined from a conventional droop slope of frequency versus measured active power output. As shown in Figure 8, the general operations of a generic VSM are the following:  
* The voltages at point of common coupling (PCC) between the output of VSM and grid are measured to compute the phase currents of the VISMA in real-time. * The computed currents are utilized as reference currents for a current-controlled inverter. * The pulse width modulation (PWM) modulator drives the inverter. The active power control loop regulates frequency while the reactive power control loop regulates voltage amplitude. When the current tracking error is small, the inverter behaves as an SM. This behavior fulfills the operation of a proposed VISMA. After the first proposal, there are other topologies that emulate VI for power electronic converters, HVDC, battery storage system (BSS), microgrid, and other applications. The core concept is still the same with different control parameters, approaches, designs, mathematical models, and implementations. The model of an SM has several orders, from first- to seventh-order models. The voltages at the PCC with the grid are measured. In real-time, the measured voltages at PCC are computed to calculate the phase currents of the VISMA. The computed currents values are the reference currents for the inverter to enable the operation of an inverter as a grid-connected current source. The behavior of VISMA depends on the current tracking error. If the error is small, it behaves like a synchronous machine. If the error is large, the behavior changes. VISMA implements a control system to integrate a three-phase inverter with SG characteristics. VISMA behaves as an SG to imitate the advantages and merits of an SG. VISMA algorithm implements [PERSON]'s coordinate reference frame by applying dq transformation to the synchronous machine mathematical model. In general, VISMA and VSM have the same control configuration, which emulates VI by using swing equation. By comparison, the synchronverter has some differences.